cpb-us-e1.wpmucdn.com · A BRIEF ODUCTION INTR TO OID TR MA Y THEOR JOSEPH E. BONIN 1. oduction Intr y Man t di eren topics in mathematics, ranging from applied sub jects h suc as

A BRIEF INTRODUCTION TO MATROID THEORY

JOSEPH E. BONIN

1. Introdu tion

Many di�erent topi s in mathemati s, ranging from applied subje ts su h as

optimization to pure areas su h as arrangements of hyperplanes, naturally lead to

matroids. The approa h we will explore is matroid theory as an abstra tion of

aÆne and proje tive geometry. Therefore the �rst several se tions will survey some

elementary, although perhaps unfamiliar, aspe ts of geometry. We will dis uss

aÆne and proje tive geometry in general, although our main interest will be in

�nite geometries.

While aÆne geometry has been studied, in varying degrees of generality, for thou-

sands of years, and proje tive geometry grew out of investigations into perspe tive

during the Renaissan e, hie y by Girard Desargues (1591{1661), the study of �nite

geometries largely began in the 1930's and 1940's with the work of su h mathemati-

ians as Marshall Hall, Jr., Ri hard H. Bru k, and Herbert Ryser. We will fo us

on basi aspe ts of aÆne and proje tive geometries; this part of our survey is not

meant as an introdu tion to urrent resear h in �nite geometry, whi h ontinues to

be a subje t of intense resear h a tivity.

At its most fundamental level, geometry is on erned with su h simple notions

as points, lines, planes, and their higher-dimensional ounterparts. We an onsider

su h on epts even if we do not have a notion of distan e (whi h would be made

pre ise by a metri ). This is exa tly what we will onsider: non-metri geometry.

Thus, we will not have angles, urvature, or even the notion of \between". At �rst

it may seem that su h a minimalisti version of geometry would be too limited to

be interesting, but this is far from the ase.

2. Affine Geometries

We start with aÆne geometry, whi h abstra ts the familiar properties of R

n

.

De�nition 2.1. An aÆne geometry is a set S of points and two olle tions of

subsets of S, the set of lines and the set of planes, subje t to these axioms:

(A1) ea h pair A;B of distin t points is ontained in a unique line, whi h is denoted

`(A;B),

(A2) ea h triple of non ollinear points is ontained in a unique plane,

(A3) if P is a point not in a line `, then there is a unique line `

�

with P in `

�

and

` parallel to `

�

(parallel lines are oplanar and disjoint),

(A4) the relation \parallel or equal" is an equivalen e relation, and

(A5) ea h line has at least two points.

Axiom (A3) is the parallel postulate. Note that the re exive and symmetri

properties automati ally hold for the relation in axiom (A4); thus, the only issue is

the transitive property. Axiom (A5) ex ludes ertain degenerate ases.

First posted May 20, 2000; updated April 3, 2001.

1

2 JOSEPH E. BONIN

The interpretation of these axioms in R

n

is familiar: points are n-tuples in R

n

,

lines are aÆne lines in R

n

(i.e., lines that need not go through the origin), and

planes are aÆne planes in R

n

(i.e., planes that need not go through the origin).

It is useful to think of these from a slightly more algebrai perspe tive: points

are translations (or osets) of the zero subspa e, lines are the translations of the 1-

dimensional (linear) subspa es, and planes are the translations of the 2-dimensional

(linear) subspa es.

To get more examples of aÆne geometries, we ould repla e R by the elements

of any division ring F (a stru ture that satis�es all the axioms of a �eld ex ept

perhaps the ommutative law of multipli ation). All basi results of linear algebra

(in parti ular, all theorems about dimension and subspa es) are valid for ve tor

spa es over arbitrary division rings. The ase of most interest for us will be that

in whi h F is a �nite �eld, the Galois �eld GF (q) for some prime power q. If q is

prime, this �eld is Z

q

, the integers 0; 1; : : : ; q � 1 with arithmeti modulo q.

Thus, let F be a division ring. Let AG(n; F ) be the aÆne geometry with the

following points, lines, and planes: the points are the n-tuples of F

n

, i.e., the trans-

lations of the zero subspa e of F

n

, the lines are the translations of the 1-dimensional

subspa es of F

n

, and the planes are the translations of the 2-dimensional subspa es

of F

n

. Of ourse, F

n

ould be repla ed by any n-dimensional ve tor spa e over

F . Verifying axioms (A1){(A5) is straightforward; for instan e, the unique line

that ontains ve tors A and B is fA + �(A � B) j � 2 Fg, the translation of the

1-dimensional subspa e f�(A�B) j � 2 Fg by A.

Consider AG(2; 3), the aÆne plane over GF (3), the �eld of three elements (i.e.,

f0; 1; 2g under arithmeti modulo 3). (The notation AG(n;GF (q)) is shortened to

AG(n; q).) There are nine points, (0; 0), (1; 0), (2; 0), (0; 1), (1; 1), (2; 1), (0; 2),

(1; 2), and (2; 2). Note that f(0; 0); (0; 1); (0; 2)g is a subspa e; its translations

are f(1; 0); (1; 1); (1; 2)g and f(2; 0); (2; 1); (2; 2)g. Note that these three lines are

parallel. In this manner, we get three other equivalen e lasses of parallel lines,

f(0; 0); (1; 0); (2; 0)g and its osets f(0; 1); (1; 1); (2; 1)g and f(0; 2); (1; 2); (2; 2)g;

f(0; 0); (2; 1); (1; 2)g and its osets f(1; 0); (0; 1); (2; 2)g and f(2; 0); (1; 1); (0; 2)g;

and

f(0; 0); (1; 1); (2; 2)g and its osets f(1; 0); (2; 1); (0; 2)g and f(2; 0); (0; 1); (1; 2)g:

These sets are shown in Figure 1. Together these give the points and lines of

AG(2; 3) as shown in Figure 2.

The aÆne geometry AG(n; q) has q

n

points sin e this is the number of n-tuples

over the q-element �eld GF (q). There are q points on ea h line of AG(n; q) sin e

the lines are the translations fv + �u j � 2 GF (q)g of the 1-dimensional subspa es

f�u j � 2 GF (q)g. Likewise there are q

2

points in ea h plane of AG(n; q) sin e

the planes are the translations fv + �u+ �w j �; � 2 GF (q)g of the 2-dimensional

subspa es f�u+ �w j �; � 2 GF (q)g.

The examples of aÆne geometries that we have seen so far suggest basi stru -

tural features of aÆne geometries. In parti ular, there is a bije tion between the

points on any two lines of an aÆne geometry, and between the points in any two

planes. The proofs are good exer ises.

If all lines in an aÆne geometry are �nite and have exa tly q points, we say that

q is the order of the aÆne geometry. Thus, AG(n; q) has order q.

A BRIEF INTRODUCTION TO MATROID THEORY 3

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10

20

01

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Figure 1. The four families of parallel lines in AG(2; 3). Sub-

spa es are depi ted with heavy lines; the osets are dotted.

00

10

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01

11

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02

12

22

Figure 2. The aÆne plane AG(2,3).

We have mentioned points, lines, and planes. There may be aÆne subspa es of

higher dimension. Must we list these separately or are they somehow determined

by the points, lines, and planes? The notion of a at, whi h is the geometri term

for subspa e, shows how to apture all subspa es using only points and lines.

4 JOSEPH E. BONIN

X X

Line-closedNot line-closed

Figure 3

Figure 4

De�nition 2.2. In an aÆne geometry in whi h there are at least three points on

ea h line, a at is a subset X of the set of points that satis�es the line- losure

ondition: if A;B 2 X, then `(A;B) � X.

Figure 3 gives a generi pi ture of the ondition of line- losure. Figure 4 shows

a subset (the ir led points) of AG(2; 3) that is not a at.

Note that De�nition 2.2 uses only lines. Apart from aÆne geometries in whi h

all lines have exa tly two points, the only role planes have is in allowing us to talk

about parallel lines. (Re all that parallel lines are oplanar lines that are disjoint,

so this relies on knowing what planes are.) Planes have a more important role in

aÆne geometries in whi h lines have exa tly two points, but we will gloss over this.

The ats of an aÆne geometry on a set S of points have the following important

properties that will be abstra ted in the de�nition of a matroid. (Sin e the �nite

ase is ultimately our main interest, we are impli itly assuming S is �nite; the

general ase uses the same ideas and only slightly more umbersome notation for

the third property.)

(F1) The set S is a at.

(F2) The interse tion of any olle tion of ats is a at.


(F3) If X is a at and X

1

; X

2

; : : : ; X

t

are the ats that overX (i.e., X is a proper

subset of X

i

, denoted X � X

i

, and there is no at Y with X � Y � X

i

),

then the di�eren es X

1

�X;X

2

�X; : : : ;X

t

�X partition S �X .

Properties (F1) and (F2) are immediate from De�nition 2.2. Property (F3) is more

ompli ated and would require a distra ting digression to prove in this setting. We

will justify it indire tly later in two ways: in the �rst justi� ation, we will state

that our model AG(n; F ) overs almost all instan es of aÆne geometries, and one

an he k (F3) dire tly for this model; in the se ond justi� ation, we will prove the

analogous property for proje tive geometries and ite a onne tion that allows us

to translate between the two settings.

Rather than proving (F3) here, we fo us on what it is saying. It is the natural

generalization of the observation that given a line and a point not on the line, the

point and the line determine a unique plane; in other words, the planes through a

line partition the set of points that are not on that line. This spe ial ase of (F3)

follows easily from axioms (A2) and (A3).

One an easily show that the ats of AG(n; F ) are the translations of the linear

subspa es of F

n

. Note that property (F3) holds for an i-dimensional linear subspa e

X of F

n

: this is simply saying that ea h ve tor u not in X is in a unique (i + 1)-

dimensional subspa e that ontains X , namely span(X [ fug). To get the general

ase for AG(n; F ) from this, just translate: pi k x in a at X ; the translation

fy � x j y 2 Xg is a linear subspa e, so the ats overing it partition the set of

points that are not in X ; the translations of these ats by x are the overs of X

and the partitioning property is preserved.

Property (F2) has an important onsequen e: Given a set T of points of an aÆne

geometry, there is a unique smallest at that ontains T , namely the interse tion

of all ats that ontain T . By property (F1), this is the interse tion of a nonempty

olle tion of sets.

De�nition 2.3. The losure l(T ) of a set T of points in an aÆne geometry is

given by

l(T ) =

\

ats X

with T�X

X:

One an think of the losure l(T ) of T as the at spanned by T . In any aÆne

geometry, l(;) = ;; in parti ular, ; is a at.

The losure of the set of ir led points in Figure 4 is the entire plane, AG(2; 3).

The notion of losure allows us to de�ne the rank of a at.

De�nition 2.4. The rank r(X) of a at X in an aÆne geometry is given by

r(X) = minfjT j : T � X and l(T ) = Xg:

The rank of a at aptures how many points it takes to determine the at. For

instan e, it takes two points to determine a line, so lines have rank two; likewise,

it takes three points to determine a plane, so planes have rank three. The rank of

an aÆne geometry is the rank of its ground set.

Rank is losely linked to dimension. One an show that in AG(n; F ), the aÆne

geometry onstru ted from F

n

, the ats of rank i are pre isely the osets of the

(i � 1)-dimensional subspa es of F

n

. In parti ular, F

n

, and hen e AG(n; F ), has

rank n + 1. Be ause of this we prefer to shift the notation; we will fo us on

AG(n� 1; F ), the rank-n aÆne geometry that is onstru ted from F

n�1

.

6 JOSEPH E. BONIN

It is natural to ask: Are there aÆne geometries in addition to the examples

AG(n�1; F ) we onstru ted from division rings? The following important theorem

says that all aÆne geometries of rank 4 and greater are of the form AG(n � 1; F ).

(This is one of several di�erent theorems that various authors ite as the funda-

mental theorem of aÆne geometry.)

Theorem 2.5 (The Fundamental Theorem of AÆne Geometry). Ea h aÆne ge-

ometry of rank n, where n � 4, is isomorphi to AG(n � 1; F ) for some division

ring F .

Thus, apart from aÆne geometries of low rank (spe i� ally, aÆne lines and aÆne

planes), aÆne geometry is pre isely the study of the osets of the subspa es of a

ve tor spa e over a division ring.

Theorem 2.5 has evolved over time; it may not be possible to attribute it to any

one person.

Rank two aÆne geometries, i.e., aÆne lines, obviously are not very interesting

and have no meaningful orresponden e with any algebrai stru ture. Rank three

aÆne geometries, i.e., aÆne planes, are the subje t of intense resear h, largely

through the orresponding proje tive planes. We end this se tion with a very brief

mention of some intriguing aspe ts of aÆne planes. For more about this fas inating

area, see, e.g., [24℄.

There are many algebrai stru tures (e.g., near �elds, Veblen-Wedderburn sys-

tems) that are mu h less onstrained than are division rings that, nonetheless, have

enough stru ture to give rise to aÆne planes.

Re all that the order of a �nite aÆne plane is the number of points in ea h line.

Thus, for a prime power q, the aÆne plane AG(2; q) has order q. It is known that

for every proper prime power q = p

k

(\proper" means k > 1) apart from q = 4 and

q = 8, there are at least two nonisomorphi aÆne planes of order q. The following

question has resisted all atta ks for well over half a entury.

Open Problem 2.6. Must the order of a �nite aÆne plane be a prime power?

The Bru k Ryser theorem is a powerful tool for showing that a parti ular number

is not the order of any aÆne plane; however this theorem addresses only numbers

that are ongruent to 1 or 2 modulo 4 and the impli ation is valid in only one

dire tion.

Theorem 2.7 (Bru k and Ryser, 1949). If q is ongruent to 1 or 2 modulo 4 and

there is an aÆne plane of order q, then q is a sum of two squares.

This theorem rules out aÆne planes of order 6, for instan e, sin e 6 is ongruent

to 2 modulo 4 but is not a sum of two squares. (Tarry's proof on the non-existen e

of 6 by 6 orthogonal Latin squares, given around 1910, also shows that aÆne planes

of order 6 do not exist.) Note that 10 is ongruent to 2 modulo 4 and 10 is 3

2

+1

2

.

However, in 1988 it was shown that there is no aÆne plane of order 10. (The

proof involved a massive omputer sear h.) Thus, the onverse of the Bru k Ryser

theorem is false. The smallest positive integer for whi h we urrently do not know

whether there is an aÆne plane of that order is 12.

The most important things to remember from this se tion are properties (F1){

(F3) sin e abstra ting these gives the de�nition of a matroid. The aÆne geometries

AG(n�1; q) will also play an important role in what follows, so we lose this se tion

by summarizing the basi properties of these geometries.


AG(n� 1; q)

AG(n� 1; q) has rank n.

AG(n� 1; q) has q

n�1

points.

Lines of AG(n� 1; q) have q points.

For i > 0, rank-i ats of AG(n� 1; q) have q

i�1

points.

3. Proje tive Geometries

A ording to Theorem 2.5, for ranks ex eeding three aÆne geometry is the

study of the osets of the subspa es of a ve tor spa e over a division ring. Sin e the

subspa es themselves, rather than their osets, are the main fo us of linear algebra,

you might wonder what geometri stru tures arise from subspa es. Subspa es give

rise to proje tive geometries.

Let's start with (GF (2))

3

, the 3-dimensional ve tor spa e over the two-element

�eld GF (2). There are eight ve tors in this ve tor spa e:

(0; 0; 0); (0; 0; 1); (0; 1; 0); (0; 1; 1); (1; 0; 0); (1; 0; 1); (1; 1; 0); (1; 1; 1):

There is a unique 0-dimensional subspa e, f(0; 0; 0)g. There are seven 1-dimensional

subspa es, orresponding to the seven nonzero ve tors in (GF (2))

3

:

f(0; 0; 0); (0; 0; 1)g; f(0; 0; 0); (0; 1; 0)g; f(0; 0; 0); (0; 1; 1)g; f(0; 0; 0); (1; 0; 0)g;

f(0; 0; 0); (1; 0; 1)g; f(0; 0; 0); (1; 1; 0)g; and f(0; 0; 0); (1; 1; 1)g:

There are seven 2-dimensional subspa es:

f(0; 0; 0); (0; 0; 1); (0; 1; 0); (0; 1; 1)g;

f(0; 0; 0); (0; 0; 1); (1; 0; 0); (1; 0; 1)g;

f(0; 0; 0); (0; 0; 1); (1; 1; 0); (1; 1; 1)g;

f(0; 0; 0); (0; 1; 0); (1; 0; 0); (1; 1; 0)g;

f(0; 0; 0); (0; 1; 0); (1; 0; 1); (1; 1; 1)g;

f(0; 0; 0); (0; 1; 1); (1; 0; 0); (1; 1; 1)g;

f(0; 0; 0); (1; 0; 1); (1; 1; 0); (0; 1; 1)g:

Of ourse, the eight ve tors of (GF (2))

3

form the unique 3-dimensional subspa e

of this ve tor spa e.

We an draw a diagram of these subspa es, just as we drew diagrams for aÆne

geometries. Sin e (0; 0; 0) is in all subspa es, we lose no information by suppressing

it in the diagrams. The resulting diagram is given in Figure 5.

Noti e that this geometry has some properties quite unlike those of aÆne ge-

ometries. In parti ular, every pair of lines has a point of interse tion: there are

no parallel lines. This property will hold in general for oplanar lines if we take

as points the 1-dimensional subspa es of a ve tor spa e, as lines the 2-dimensional

subspa es of the ve tor spa e, and as planes the 3-dimensional subspa es of the

ve tor spa e. This follows from the familiar dimension theorem of linear algebra:

dim(U

1

) + dim(U

2

) = dim(U

1

+ U

2

) + dim(U

1

\ U

2

):

8 JOSEPH E. BONIN

100

010 011 001

110 111 101

Figure 5. The Fano Plane, F

7

or PG(2; 2).

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

A

B

C

D

Figure 6. The on�guration in the Pas h axiom.

Letting U

1

and U

2

be lines `

1

and `

2

(i.e., 2-dimensional subspa es) in a plane (i.e.,

a 3-dimensional subspa e), we have

dim(`

1

)

| {z }

2

+dim(`

2

)

| {z }

2

= dim(`

1

+ `

2

)

| {z }

3

+dim(`

1

\ `

2

);

so dim(`

1

\ `

2

) is 1, that is, `

1

\ `

2

is a 1-dimensional subspa e, or a point.

With this motivation, we an now de�ne proje tive geometries.

De�nition 3.1. A proje tive geometry is a set S of points and a olle tion of

subsets of S, the set of lines, subje t to these axioms:

(P1) ea h pair A;B of distin t points is ontained in a unique line, whi h is denoted

`(A;B),

(P2) if A;B;C, and D are distin t points for whi h `(A;B) \ `(C;D) 6= ;, then

`(A;C) \ `(B;D) 6= ;, and

(P3) ea h line ontains at least three points.

Axiom (P2) is the Pas h axiom. It is a way of saying oplanar lines interse t

without mentioning planes; intuitively (and in a sense we ould easily make pre ise),

sin e `(A;B) \ `(C;D) 6= ;, all four points A;B;C, and D lie in a plane, so the

lines `(A;C) and `(B;D) are therefore oplanar and so should interse t nontrivially.

This is illustrated in Figure 6.

Noti e that the axiom system for proje tive geometry is onsiderably simpler

than for aÆne geometry; there are three axioms, rather than �ve, and we mention

only points and lines, not planes. This is typi al: proje tive geometry is simpler


than aÆne geometry, even though, as we will see, it en ompasses aÆne geometry.

The reason is that, as we will make pre ise later, proje tive geometry is the natural

ompletion of aÆne geometry.

Observe also that if the proje tive geometry is a plane, we have a beautiful

symmetry in the axioms: any pair of points spans a unique line and any pair of

lines interse ts in a unique point. This has a striking resemblan e to duality in

linear algebra; this is no a ident.

It should ome as no surprise now that we get a proje tive geometry out of every

ve tor spa e over a division ring; simply take the points to be the 1-dimensional

subspa es and the lines to be the 2-dimensional subspa es, viewed as sets of the

points. Alternatively, pi k pre isely one nonzero representative ve tor out of ea h

1-dimensional subspa e, and let the points be the hosen ve tors and the lines be

the sets of these hosen ve tors that are in 2-dimensional subspa es. The proje -

tive geometry onstru ted in this manner from (GF (q))

n

is denoted PG(n� 1; q).

More generally, the proje tive geometry onstru ted in this manner from an n-

dimensional ve tor spa e over a division ring F is denoted PG(n� 1; F ).

Note that this is pre isely how the real proje tive plane PG(2;R) is formed from

R

3

in topology and geometry. The standard onstru tions of the real proje tive

plane take one of the following three equivalent approa hes, ea h of whi h illustrates

what is stated in the last paragraph. Take as the points of PG(2;R) the lines of

R

3

through the origin, with the origin deleted; ea h plane of R

3

through the origin

gives rise to the line of PG(2;R) that onsists of all points of PG(2;R) that lie in

this plane, and all lines of PG(2;R) are of this form. Alternatively, hoose as the

points of PG(2;R) pre isely one out of ea h pair of antipodal points on the unit

sphere in R

3

(for example, the points on the upper half sphere, with half of the

edge in luded), and let the lines of PG(2;R) be the sets of these points of PG(2;R)

that are interse tions of the set of points of PG(2;R) with planes of R

3

through the

origin. Alternatively, take as the points of PG(2;R) the pairs of antipodal points

on the unit sphere in R

3

, and let the lines of PG(2;R) be the sets of these points

of PG(2;R) that are interse tions of the set of points of PG(2;R) with planes of

R

3

through the origin; �nally, identify antipodal points.

As with aÆne geometries, we will fo us on �nite proje tive geometries. We

illustrate the onstru tion above with a se ond �nite example (Figure 5 being the

�rst example, although too small to illustrate the hoi e of representative ve tors):

Figure 7 shows the proje tive plane PG(2; 3) formed from the 3-dimensional ve tor

spa e (GF (3))

3

over the �eld GF (3). In this diagram, the point (1; 1; 2) represents

the 1-dimensional subspa e f(0; 0; 0); (1; 1; 2); (2; 2; 1)g; the line

f(0; 1; 2); (1; 2; 1); (1; 1; 2); (1; 0; 0)g

represents the 2-dimensional subspa e

f(0; 0; 0); (0; 1; 2); (0; 2; 1); (1; 2; 1); (2; 1; 2); (1; 1; 2); (2; 2; 1); (1; 0; 0); (2; 0; 0)g:

Noti e that while AG(2; 3) has three point on ea h line, PG(2; 3) has four points

on ea h line.

The geometry PG(n�1; q) has (q

n

�1)=(q�1) points sin e the points orrespond

to the 1-dimensional subspa es of (GF (q))

n

, and there are q

n

�1 nonzero ve tors in

(GF (q))

n

with q�1 nonzero ve tors in ea h 1-dimensional subspa e. Alternatively,

by hoosing as a representative ve tor of a subspa e P = hvi the unique multiple

of v that has a 1 in the �rst nonzero position (as shown in Figure 7), we see that

10 JOSEPH E. BONIN

001

012

011

010

100 101 102

110 111 112

120 121 122

Figure 7. The proje tive plane PG(2; 3).

there are (q

n

�1)=(q�1), or q

n�1

+q

n�2

+ � � �+q+1, points sin e the �rst nonzero

position ould be the �rst (leaving q

n�1

ways to �ll in the other n� 1 entries) or

the se ond (leaving q

n�2

ways to �ll in the other n� 2 entries), and so on.

The notion of a at is essentially the same as in aÆne geometries.

De�nition 3.2. A at in a proje tive geometry is a set X of points in the geometry

that satis�es the line- losure ondition: if A;B 2 X, then `(A;B) � X.

In keeping with our observation that proje tive geometry is simpler than aÆne

geometry, there is no ex eptional ase like the ase of two-point lines in aÆne

geometries. Observe that De�nition 3.2 is simply the geometri formulation of the

familiar algebrai de�nition of a linear subspa e. Thus, the ats in the proje tive

geometry arising from a division ring are pre isely the subspa es (or the sets of the

representative ve tors in the subspa es).

The ats of a proje tive geometry on a set S of points have the following im-

portant properties that we also saw for the ats of an aÆne geometry and that we

will see again in the de�nition of a matroid. (As was the ase for aÆne geometries,

we are impli itly assuming S is �nite sin e this is ultimately the ase of greatest

interest.)

(F1) The set S is a at.

(F2) The interse tion of any olle tion of ats is a at.

(F3) IfX is a at andX

1

; X

2

; : : : ; X

t

are the ats that overX , then the di�eren es

X

1

�X;X

2

�X; : : : ;X

t

�X partition S �X .

Properties (F1) and (F2) are immediate from De�nition 3.2.

Given that the ats in the proje tive geometry arising from a ve tor spa e are

(sets of representatives from) the subspa es, property (F3) has this familiar inter-

pretation: for an i-dimensional linear subspa e X of a ve tor spa e, ea h ve tor

u not in X is in a unique (i + 1)-dimensional subspa e that ontains X , namely

span(X [ fug).

It is easy to justify property (F3) in general (again, we fo us on the ase in whi h

S is �nite). Note that property (F3) trivially holds if X is the empty set (whi h


XC' D'

C DE

E'

P

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Figure 8

is a at by De�nition 3.2), for then the overing ats X

1

; X

2

; : : : ; X

t

are just the

singleton sets of points (whi h are also ats). So assume that X is a nonempty at.

The key to proving property (F3) is to identify the at that overs X and ontains

a point P that is not in X . We laim that the at that overs X and ontains a

point P that is not in X is

X

P

=

[

A2X

`(A;P );(1)

that is, the set of all points that are ollinear with P and a point of X . Clearly

any at that ontains X and P ontains all of X

P

, so all we need to show is that

X

P

satis�es the line- losure ondition that de�nes ats. Toward this end assume

C and D are in X

P

. By the de�nition of X

P

, there are points C

0

and D

0

in

X with C 2 `(C

0

; P ) and D 2 `(D

0

; P ). If C

0

= D

0

, then `(C;D) = `(P;C

0

),

hen e `(C;D) � X

P

as desired, so assume C

0

6= D

0

. (See Figure 8.) Let E be in

`(C;D); we need to show that E is in X

P

, i.e., that there is a point E

0

in X with

E 2 `(E

0

; P ). Now (skipping the appli ations of the Pas h axiom that fully justify

this) note that sin e `(C

0

; D

0

) and `(E;P ) are oplanar lines, they interse t at some

point E

0

, whi h is ne essarily in X sin e X is a at that ontains both C

0

and D

0

.

Thus E is indeed in X

P

, so X

P

is a at. Now to verify property (F3), note that we

have shown that ea h point P not in X is in a smallest at X

P

that ontains X and

P and equation (1) gives an expli it expression for this at. Assume that for two

points P and Q, the sets X

P

�X and X

Q

�X are not disjoint, say both ontain

the point R. Sin e X

P

is a at that ontains X and R, we have X

R

� X

P

; on the

other hand, sin e R is in X

P

, we know that R and P are ollinear with a point of

X , so P is in X

R

, and so X

P

� X

R

; therefore X

P

= X

R

. Similarly, X

Q

= X

R

.

Therefore X

P

= X

Q

. Thus we have shown that the only ats that over X and

interse t in a proper superset of X are identi al, whi h is the required partitioning

property.

Several more ideas we saw for aÆne geometries arry over to proje tive geome-

tries. In parti ular, as a onsequen e of property (F2), for ea h set T of points

in a proje tive geometry there is a unique smallest at ontaining T , namely the

interse tion of all ats that ontain T . By Property (F1), this is the interse tion of

a nonempty olle tion of sets.

12 JOSEPH E. BONIN

Figure 9. Remove the points of a hyperplane (dotted) of PG(2; 3)

to get AG(2; 3).

De�nition 3.3. The losure l(T ) of a set T of points in a proje tive geometry is

given by

l(T ) =

\

ats X

with T�X

X:

Again, losure gives rise to the notion of rank.

De�nition 3.4. The rank r(X) of a at X in a proje tive geometry is given by

r(X) = minfjT j : T � X and l(T ) = Xg:

In the proje tive geometry arising from a ve tor spa e, the ats of rank i are (the

sets of representatives of) the i-dimensional subspa es. Thus, rank is the proje tive

geometry ounterpart of dimension in linear algebra. Note that PG(n � 1; q) has

rank n, and the rank-i ats of PG(n�1; q) have (q

i

�1)=(q�1) points. In parti ular,

lines of PG(n� 1; q) have (q

2

� 1)=(q � 1), or q + 1, points.

In linear algebra, the subspa es of dimension n � 1 in an n-dimensional ve tor

spa es are alled the hyperplanes. Analogously, the ats of rank n�1 in a proje tive

geometry of rank n are alled hyperplanes.

We have alluded to the fa t that proje tive geometry is the natural ompletion

of aÆne geometry. This is exempli�ed by the onne tion between AG(2; 3) and

PG(2; 3) suggested in Figure 9. The pre ise formulation of this is the following

theorem, whi h is easy to prove.

Theorem 3.5. Let H be a hyperplane of a proje tive geometry on the set S of

points and let L and P be the set of lines and the set of planes (i.e., ats of rank

3) of the geometry. We get an aÆne geometry with S

0

= S�H as the set of points

by taking as the set of lines

L

0

= f` \ S

0

j ` 2 L with ` 6� Hg

and as the set of planes

P

0

= f� \ S

0

j � 2 P with � 6� Hg:

Conversely, assume that L

0

is the set of lines of an aÆne geometry on a set S

0

,

and that P

0

is the set of planes of this geometry. Let fL

0

i

j i 2 Ig be the set of

equivalen e lasses of parallel lines. With ea h equivalen e lass L

0

i

, let A

i

be a point

not in S

0

. Let S be the set S

0

[fA

i

j i 2 Ig. With ea h line `

0

of L

0

, let ` be `

0

[fA

i

g


where `

0

is in L

0

i

. With ea h plane � of P

0

, let `

�

be fA

i

j � ontains a line in L

0

i

g.

Then S together with the set

L = f` j `

0

2 L

0

g [ f`

�

j � 2 P

0

g

of lines is a proje tive geometry.

Thus, aÆne and proje tive geometry are intimately linked. To get an aÆne

geometry from a proje tive geometry, remove all points in a hyperplane and onsider

the indu ed sets of lines and planes. (The operation of deletion that this illustrates

is a basi matroid operation.) To get a proje tive geometry from an aÆne geometry,

add one point to all lines in ea h equivalen e lass of parallel lines and let the lines

of new points orrespond to the planes of the aÆne geometry.

As there are aÆne planes that do not arise from our onstru tion based on ve tor

spa es over a division ring, there are proje tive planes that do not arise from our

onstru tion based on ve tor spa es over a division ring. Su h planes may have

some rather unexpe ted properties. For instan e, with a proje tive plane that does

not arise from a division ring, it may be possible to remove di�erent lines and get

nonisomorphi aÆne planes. Thus, while ea h aÆne plane has a unique ompletion

to a proje tive plane given by the onstru tion in Theorem 3.5, many nonisomorphi

aÆne planes an have the same ompletion to a proje tive plane.

Given Theorems 2.5 and 3.5, the next theorem should ome as no surprise.

Theorem 3.6 (The Fundamental Theorem of Proje tive Geometry). Every pro-

je tive geometry of rank n, where n � 4, is isomorphi to PG(n � 1; F ) for some

division ring F .

Thus, apart from planes, proje tive geometry is the study of the subspa es of a

ve tor spa e over a division ring.

Given that proje tive geometry is simpler than aÆne geometry, although the

two subje ts are equivalent in the sense made pre ise in Theorem 3.5, the problems

mentioned at the end of Se tion 2 are typi ally studied for proje tive planes rather

than aÆne planes.

As in the last se tion, the most important things to remember from this se tion

are properties (F1){(F3). The proje tive geometries PG(n� 1; q) will also play an

important role in what follows, so we lose this se tion by summarizing the basi

properties of these geometries.

PG(n� 1; q)

PG(n� 1; q) has rank n.

PG(n� 1; q) has (q

n

� 1)=(q � 1), or q

n�1

+ q

n�2

+ � � �+ q + 1, points.

Lines of PG(n� 1; q) have q + 1 points.

Rank-i ats of PG(n� 1; q) have (q

i

� 1)=(q � 1) points.

4. Coordinates

To motivate some of the topi s we will see in matroid theory, it is useful to

sket h some of the elements that go into the proofs of Theorems 2.5 and 3.6. These

ideas go ba k well over a hundred years; they are used in Hilbert's book [23℄, whi h

�rst appeared in 1899, although they were known well before then. For a omplete,

elementary presentation from a modern perspe tive, see [2℄.

14 JOSEPH E. BONIN

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PP

P

OA

BC

A'

B'

C'

1

2

3

Figure 10. Desargues' on�guration.

The problem of realizing the ats of a proje tive geometry as the subspa es of

a ve tor spa e over a division ring is the problem of oordinatizing the geometry.

This is equivalent to oordinatizing the orresponding aÆne geometry. We will shift

freely between aÆne and proje tive geometry.

We mentioned that not all proje tive planes arise from division rings. It is

natural to ask: Is it possible to hara terize the proje tive planes that arise from

division rings? This is pre isely what Desargues' theorem does.

Theorem 4.1. A proje tive plane is isomorphi to a proje tive plane arising from

a division ring if and only if it satis�es the following ondition:

Desargues' Theorem. Given any triples A;B;C and A

0

; B

0

; C

0

of

non ollinear points, if the lines `(A;A

0

), `(B;B

0

), and `(C;C

0

) are on-

urrent (i.e., meet in a point), then the three points `(A;B)\ `(A

0

; B

0

),

`(A;C) \ `(A

0

; C

0

), and `(B;C) \ `(B

0

; C

0

) are ollinear.

Thus, Desargues' theorem (whi h, in some form, dates ba k to the 1600's) is,

for us, not a theorem; it is a ondition, or axiom, that hara terizes the proje tive

planes that arise from division rings. (With the earlier, more limited view of geom-

etry, this was indeed a theorem sin e the only geometries onsidered satis�ed this

ondition.)

Brie y, Desargues' theorem says that two triangles that are perspe tive from

a point are perspe tive from a line. What this means is the following. The two

triangles A;B;C and A

0

; B

0

; C

0

being perspe tive from the point O (see Figure 10)

has this physi al interpretation in Eu lidean spa es: if one pla es one's eye at point

O, the two triangles A;B;C and A

0

; B

0

; C

0

line up exa tly. The triangles being

perspe tive from a line is the dual notion: instead of saying \the lines through or-

responding points are on urrent," inter hange points and lines to get \the points

at the interse tion of orresponding lines are ollinear."

Although Theorem 4.1 deals with planes, Desargues' Theorem an be interpreted

as des ribing a on�guration in a plane or in rank 4. Note that if this is a on�gu-

ration in rank 4, then P

1

; P

2

, and P

3

are in the plane l(fA;B;Cg) and in the plane

l(fA

0

; B

0

; C

0

g); sin e the interse tion of ats is a at by (F2), the points P

1

; P

2

,


A

B

C

A'

B'

C'xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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Figure 11. One aÆne version of Desargues' on�guration.

and P

3

, being in l(fA;B;Cg) \ l(fA

0

; B

0

; C

0

g), must indeed be on a line. This

begins to suggest why Theorems 2.5 and 3.6 apply when the rank is at least four.

It is an easy exer ise to show that Desargues' Theorem is equivalent to its on-

verse.

Theorem 4.2. A proje tive plane satis�es Desargues' Theorem if and only if it

satis�es the following ondition:

Given any triples A;B;C and A

0

; B

0

; C

0

of non ollinear points, if the

points `(A;B)\`(A

0

; B

0

), `(A;C)\`(A

0

; C

0

), and `(B;C)\`(B

0

; C

0

) are

ollinear, then the lines `(A;A

0

), `(B;B

0

), and `(C;C

0

) are on urrent.

We have mentioned several times that proje tive geometry is simpler than aÆne

geometry. This is in part be ause one proje tive on�guration su h as Desargues'

on�guration an give rise to many aÆne on�guration. For instan e, if none

of the points in Desargues' on�guration are in the hyperplane of a proje tive

geometry that is deleted to form an aÆne geometry, then we get the identi al

on�guration in the aÆne geometry. However, if O is ollinear with P

1

; P

2

, and P

3

in the proje tive geometry, and these are all in the hyperplane that is deleted, the

orresponding aÆne on�guration is given in Figure 11. The aÆne interpretation

of this on�guration is the following:

Given triangles A;B;C and A

0

; B

0

; C

0

with the lines `(A;A

0

), `(B;B

0

),

and `(C;C

0

) all parallel, if `(A;B) k `(A

0

; B

0

) and `(A;C) k `(A

0

; C

0

),

then `(B;C) k `(B

0

; C

0

).

In the Eu lidean plane, this statement is obvious: it follows immediately from

basi results on parallelograms. There are two other aÆne ases of Desargues'

on�guration (it is a good exer ise to identify them), but the two we have mentioned

are the most relevant for us.

Theorem 4.1 suggests that we should be able to de�ne the operations of addition

and multipli ation of some division ring in any aÆne plane in whi h Desargues'

theorem holds. Let's brie y sket h this so we an see the role Desargues' theorem

plays. (See Figure 12.) To add points A and B on a line on whi h a zero has

been designated, onstru t a parallel auxiliary line and hoose a point D on it.

Constru t `(0; D) and let the parallel line through A interse t the auxiliary line at

D

0

. Constru t `(B;D) and the parallel line through D

0

; where this parallel line

interse ts the original line is A+B.

16 JOSEPH E. BONIN

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0

D D'




0

D D'

A+B





(i)

(ii)

Figure 12. Addition in Desarguesian planes.

A Bxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx



0

D D'

A+Bxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx



E E'xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


Figure 13

This onstru tion has the obvious interpretation in the Eu lidean plane of using

parallel lines to translate the distan e from 0 to A up to the se ond line and then

ba k down to the original line, but starting at B rather than at 0. However the

onstru tion makes sense in any aÆne plane sin e it uses only parallel lines. The

issue is: When is this operation well-de�ned? Spe i� ally, when is this onstru tion

independent of the hoi e of the auxiliary line and the auxiliary point? This is the

role that Desargues' theorem plays.

To see that this operation is well-de�ned, onsider A + B as onstru ted from

an auxiliary point D, and onsider an auxiliary point E on a di�erent auxiliary

line. (See Figure 13.) By applying Desargues' theorem to triangles D; 0; E and

D

0

; A;E

0

, we on lude that `(E;D) is parallel to `(E

0

; D

0

). This allows us to apply

Desargues' theorem to triangles D;B;E and D

0

; A+B;E

0

, so we an on lude that

`(E;B) is parallel to `(E

0

; A+B), whi h says that omputing A+B using E gives

the same result as using D. The remaining ase is that of using two auxiliary points

on the same line parallel to the original line, but this follows from what we have


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0

DD'

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1

A Bxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx




0

DD'




1 AB

(i)

(ii)

Figure 14. Multipli ation in Desarguesian planes.

shown by omparing these with the result obtained by using an auxiliary point on

a se ond auxiliary line.

Having that this operation of addition is well-de�ned, it is an interesting series of

elementary exer ises to show that the points on a line form an abelian group under

this operation and that the groups de�ned by a di�erent hoi e of 0 or a di�erent

line are isomorphi .

The operation of multipli ation is de�ned similarly, ex ept that the auxiliary line

is one that goes through the additive identity. (See Figure 14.) Starting with the

additive identity 0, the multipli ative identity 1, and two points A and B on the line

`(0; 1), to multiply A and B hoose an auxiliary line through 0 and a point D on

this line. Constru t `(1; D) and the parallel line through A, meeting the auxiliary

line at D

0

. Next onstru t `(B;D) and the parallel line through D

0

, meeting the

original line at AB.

In the Eu lidean plane, an elementary argument using similar triangles shows

that this gives the orre t produ t. However, the de�nition uses only parallel lines

and so makes sense in any aÆne plane. The issue is whether this produ t is well-

de�ned. For this, one of the other aÆne realizations of Desargues' theorem omes

into play. (See Figure 15.) Constru t AB using D and onsider and auxiliary point

E on a di�erent auxiliary line. By applying one version of Desargues' theorem to

triangles D; 1; E and D

0

; A;E

0

, we on lude that `(E;D) is parallel to `(E

0

; D

0

).

This allows us to apply Desargues' theorem to triangles D;B;E and D

0

; AB;E

0

, so

we an on lude that `(E;B) is parallel to `(E

0

; AB), whi h says that omputing

AB using E gives the same result as using D. As we saw for addition, the general

ase follows from this.

Having that this operation of multipli ation is well-de�ned, it is another se-

quen e of elementary exer ises to show the remaining properties for a division

ring, namely that multipli ation is asso iative, that 1 is a multipli ative identity,

that ea h nonzero element has a multipli ative inverse, and that multipli ation dis-

tributes over addition. Elementary arguments also show that the resulting division

ring is, up to isomorphism, independent of the line hosen and the points 0 and 1

hosen on any line. One an then use this division ring to oordinatize the plane.

Theorem 4.1 states that Desargues' theorem hara terizes the proje tive planes

that arise from 3-dimensional ve tor spa es over division rings. What hara terizes

18 JOSEPH E. BONIN





0

DD'




1 AB



E

E'

Figure 15

1

2

3

4

5

6

Figure 16. Pappus' on�guration.

the proje tive planes that arise from 3-dimensional ve tor spa es over �elds? I.e.,

how an we geometri ally apture the axiom that multipli ation is ommutative?

This is provided by Pappus' theorem, some form of whi h dates ba k to around 300

A.D. (See Figure 16.)

Theorem 4.3. A proje tive plane is isomorphi to a proje tive plane arising from

a �eld if and only if it satis�es the following ondition:

Pappus' Theorem. If a hexagon is ins ribed alternately on two lines,

then the three points of interse tion of the opposite sides are ollinear.

Like Desargues' theorem, from the modern perspe tive Pappus' theorem is not

a theorem; it is a ondition, or axiom, that hara terizes the proje tive planes that

arise from �elds. It is an elementary exer ise to prove Theorem 4.3.

As a orollary of Theorems 4.1 and 4.3, it follows that Pappus' theorem implies

Desargues' theorem; it is an interesting elementary exer ise to prove this dire tly,

without using Theorems 4.1 and 4.3. In general, the onverse of ourse is not true

sin e there are division rings that are not �elds, but there is an important ase in

whi h the onverse holds. Re all Wedderburn's theorem: Every �nite division ring

is a �eld. Thus, for �nite proje tive planes, Desargues' theorem implies Pappus'

theorem. It seems there should be a geometri proof of this fa t that does not


simply geometri ally en ode the known algebrai proofs of Wedderburn's theorem;

presently no su h proof is known.

5. Matroids

We now generalize the two lassi al spa es, proje tive and aÆne geometries,

that we examined in the �rst part of this brief introdu tion to matroid theory.

De�nition 5.1. A matroid M is a �nite set S and a olle tion F of subsets of

S, the ats of M , su h that:

(F1) the set S is a at,

(F2) the interse tion of any olle tion of ats is a at, and

(F3) if X is a at and X

1

; X

2

; : : : ; X

t

are the ats that over X, then the di�er-

en es X

1

�X;X

2

�X; : : : ;X

t

�X partition S �X.

The set S is often alled the ground set of the matroid.

Thus, a matroid is simply a set together with a olle tion of subsets, the ats,

that satis�es the three properties we observed for the ats of proje tive and aÆne

geometries.

Matroids were introdu ed by Hassler Whitney, and, although the subje t has

subsequently developed in more dire tions than ould be imagined in the 1930's,

his founding paper [36℄ is still an ex ellent entry point into the subje t. Although

Whitney gave several di�erent equivalent formulations of a matroid, that in De�ni-

tion 5.1 is not among them. It is partly a re e tion of the large number of bran hes

of mathemati s in whi h matroids play a role that there are now over �fty di�er-

ent equivalent formulations of a matroid; about a dozen of these are frequently

used while others were devised for very spe ial purposes. When working with ma-

troids it is typi ally very useful to shift freely between several di�erent approa hes.

However, what makes for eÆ ien y and insight for resear hers in the �eld may be

onfusing to those new to the subje t, so we will restri t ourselves to the approa h

in De�nition 5.1.

In low ranks, we an draw the types of diagrams we drew for proje tive and

aÆne planes. For instan e, in the matroid F

7

in Figure 17, the ats are the empty

set, the seven points, the nine lines fD;A;Eg, fD;B;Gg, fE;C;Gg, fE;F;Bg,

fD;F;Cg, fA;F;Gg, fA;Bg, fA;Cg, and fB;Cg, and the plane onsisting of all

seven points. Note that when a line su h as fA;Bg ontains just two points, we do

not bother to draw a line between the two points sin e it is understood. Su h lines

are alled trivial lines.

To give a on rete example of axiom (F3), onsider the ats that over fAg.

These are fD;A;Eg, fA;F;Gg, fA;Bg, and fA;Cg; the resulting di�eren es,

namely, fD;Eg, fF;Gg, fBg, and fCg, indeed partition fB;C;D;E; F;Gg, the

set of points not in fAg.

While in proje tive and aÆne geometries, the ats were spe i�ed by the lines

together with the line- losure ondition, this is not the ase in arbitrary matroids;

in general we need to list all ats. Indeed, if we start with the points and lines of

F

7

and onsider the resulting line- losed sets (the sets that satisfy the line- losure

ondition), these violate axiom (F3). For instan e, the smallest line- losed set that

ontains the line fA;Bg and the point D is the entire set fA;B;C;D;E; F;Gg yet

fA;B;Cg is a line- losed set with fA;Bg � fA;B;Cg � fA;B;C;D;E; F;Gg, so

the partitioning property fails. This prompts the �rst of the open problems we will

mention.

20 JOSEPH E. BONIN

A B

C

D

E

F

G

Figure 17. The non-Fano matroid, F

7

.

Open Problem 5.2. Chara terize the olle tions of points and lines for whi h the

resulting line- losed sets are the ats of a matroid.

Although for most matroids, the line- losed sets are not the same as the ats,

when these two olle tions agree, ertain arguments be ome onsiderably simpler.

(See, e.g., [3, 9℄. See [22℄ for more on this topi .)

As in proje tive and aÆne geometries, the ats give rise to the notion of losure.

De�nition 5.3. The losure l(T ) of a set T of points in a matroid is given by

l(T ) =

\

ats X

with T�X

X:

By axiom (F1), this is the interse tion of a nonempty olle tion of sets. By

axiom (F2), l(T ) is a at.

Closure, in turn, gives rise to the notion of rank. It will be useful to generalize

this a bit, and onsider the rank of any set of points, not just ats.

De�nition 5.4. The rank r(X) of a set X in a matroid is given by

r(X) = minfjT j : T � l(X) and l(T ) = l(X)g:

By axiom (F2) there is a unique smallest at, the interse tion of all ats. (Indeed,

by axioms (F1) and (F2), the olle tion of ats forms a latti e.) This unique

smallest at has rank 0 and it is the only at of rank 0.

We borrow even more terms from proje tive and aÆne geometry: points are ats

of rank 1, lines are ats of rank 2, planes are ats of rank 3, and hyperplanes are

ats of rank n� 1 in a matroid of rank n.

Nothing in De�nition 5.1 for es the empty set to be a at; it need not be a at.

Likewise, singleton subsets of S need not be a at. (We will see examples of this

arising naturally soon.) For some purposes (e.g., some of the extremal problems we

will see later), it is useful to have the points orrespond exa tly with the elements

of S. To apture this, we introdu e the notion of a geometry.

De�nition 5.5. A ombinatorial geometry, simple matroid, or geometry is a ma-

troid in whi h the empty set and all singleton subsets of the ground set are ats.

We now turn to a onstru tion with whi h we will be able to produ e many

matroids.


1

4 7 5

2 6 3

1

4 5

2 6 3

Figure 18. The restri tion of PG(2; 2) to f1; 2; 3; 4; 5; 6g gives M(K

4

).

1

4 5

2 6 3

1

4 5

2 6 3

A

B

C

DE

F

GHI

Figure 19. The simpli� ation of a matroid.

De�nition 5.6. Let M be a matroid on the set S and let T be a subset of S. The

restri tion M jT of M to T is the matroid on T that has as ats the sets F \ T as

F ranges over the ats of M .

It is an easy exer ise to show that the ats of M jT indeed satisfy axioms (F1){

(F3) of De�nition 5.1.

For example, in the restri tion of PG(2; 2) in Figure 18, the at f1; 2; 4g of

PG(2; 2) restri ted to the subset f1; 2; 3; 4; 5; 6g gives the same at, f1; 2; 4g, while

the at f1; 6; 7g of PG(2; 2) yields the at f1; 6g in the restri tion. This restri tion

of PG(2; 2) is the matroid M(K

4

); the notation omes from a onne tion (whi h

we will not pursue) with the omplete graph K

4

.

If we fo us instead on what is being removed, the restri tion M jT is alled the

deletion Mn(S � T ). Thus, M(K

4

) is PG(2; 2)nf7g.

In Figure 19, the original matroid is one in whi h rank-1 ats are not all singletons

(so this is not a geometry); the rank-1 ats are f1; D;Eg, f2; B; Cg, f3; Fg, f4; Ag,

f5g, and f6; G;H; Ig. However, by restri ting this matroid to f1; 2; 3; 4; 5; 6g we

obtain a geometry that is intimately related to the original matroid | in some sense

it ontains the same geometri information without the multiple representatives.

What we see in this example is similar to hoosing a single representative ve tor

out of ea h 1-dimensional subspa e when forming a proje tive geometry. These are

both instan es of the simpli� ation of a matroid.

De�nition 5.7. Let M be a matroid on a set S and let T be a subset of S that

ontains no elements of the rank-0 at and pre isely one element of ea h rank-1

at. The restri tion M jT is the simpli� ation of M .

22 JOSEPH E. BONIN

Up to isomorphism, a matroid has a unique simpli� ation, so alling this the

simpli� ation is appropriate.

Using restri tion, we an now give many examples of matroids: PG(n � 1; q)

and its restri tions (or subgeometries). Note that AG(n � 1; q) is one of these

restri tions. Su h matroids, generalized mildly in the next de�nition, form a very

important lass of matroids.

De�nition 5.8. A matroid M is representable over GF (q) if the simpli� ation of

M is isomorphi to a restri tion of PG(n� 1; q) for some n.

We ould de�ne matroids representable over any �eld F , but the ase of �nite

�elds will be our hief interest.

The matroid M(K

4

) in Figure 18 is representable over GF (2), as is the matroid

in Figure 19.

Note that a matroid is representable over GF (q) if it is basi ally a subgeometry

of PG(n� 1; q), for some n, perhaps with multiple opies of points added, possibly

more opies of a point added than the number of s alar multiples of a ve tor over

GF (q). One might want to assign ve tors in (GF (q))

n

to the elements of the

matroid and represent the elements of the matroid by the olumns of a matrix over

GF (q); matri es naturally allow for repeated olumns. (Indeed, the word \matroid"

is intended to suggest a generalization of a matrix.)

The next se tion examines representable matroids in more detail.

6. Representable Matroids

It is natural to ask the following questions. Whi h matroids are representable

over a given �eld? Whi h matroids are representable over every �eld? For whi h

matroids do there exist �elds over whi h the matroids are representable? Su h

question are entral to matroid theory. They are di�erent aspe ts of the basi

question: How do we apture our motivating examples, proje tive geometries and

matroids easily obtained from them, within the lass of all matroids?

Our �rst issue is this: Are there matroids that are not representable over any

�eld? Desargues' theorem leads us to an example. (See Figure 20.) Sin e the on-

dition illustrated by the Desargues on�guration hara terizes the proje tive planes

that arise from 3-dimensional ve tor spa es over a division ring, the on�guration

in whi h all lines are the same ex ept that P

1

, P

2

, and P

3

are not ollinear is not

representable over any �eld (or any division ring). One an easily he k that this

rank-3 on�guration, the non-Desargues matroid, is indeed a matroid. Note that

this on�guration annot be interpreted as a matroid of rank 4 sin e the points P

1

,

P

2

, and P

3

would then be in the interse tion of two distin t planes, l(fA;B;Cg)

and l(fA

0

; B

0

; C

0

g), and therefore ollinear.

The same idea gives the non-Pappus matroid, a matroid that is not representable

over any �eld although it is representable over skew �elds. (See Figure 21.)

Thus, matroid theory is not limited to representable matroids; many matroids

do not arise from proje tive geometries. It is of great interest to hara terize the

matroids that are representable over a �eld. Note that any su h hara terization

must be more \sensitive" than onditions su h as Desargues' theorem or Pappus'

theorem sin e the on�gurations in these theorems are based on having ertain lines

a tually interse ting (or parallel, in the aÆne ases); there is no assumption that

oplanar lines of a matroid interse t or that a ounterpart of the parallel postulate

holds.






xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx






PP

P

OA

BC

A'

B'

C'

1

2

3











PP

P

OA

BC

A'

B'

C'

1

2

3

Figure 20. The Desargues on�guration and the non-Desargues matroid.

1

2

3

4

5

6

1

2

3

4

5

6

Figure 21. The Pappus on�guration and the non-Pappus matroid.

One immediate but very useful observation about representability is the following

theorem. (Sin e our interest is hie y in the ase of �nite �elds, we fo us on that

ase; the result holds for arbitrary �elds.)

Theorem 6.1. If a matroid M is representable over GF (q), then every restri tion

of M is representable over GF (q).

This is lear sin e the simpli� ation ofM is a restri tion of a proje tive geometry

PG(n� 1; q), so the simpli� ation of a restri tion of M is just a further restri tion

of PG(n� 1; q).

Re all that lines in PG(n� 1; q) have pre isely q +1 points. In parti ular, lines

in PG(n � 1; 2) have exa tly three points. Thus, if a matroid M has a line with

four (or more) points, then M is not representable over GF (2).

The four point line is denoted U

2;4

. (See Figure 22.) The matroid U

2;4

has four

elements, four points, and rank two. The argument in the last paragraph shows that

binary matroids (those representable over GF (2)) annot have U

2;4

as a restri tion.

The matroid U

2;4

is one of an in�nite family of very basi matroids, the uniform

matroids. Let U

n;m

be the matroid in whi h the ground set has m elements and

the ats are the entire set and all subsets of size less than n. It is easy to he k that

axioms (F1){(F3) are satis�ed. The matroid U

n;m

is alled the uniform matroid of

24 JOSEPH E. BONIN

Figure 22. U

2;4

Figure 23. U

3;5

X

A B

A' B'C D

Figure 24. U

2;4

as a proje tion of U

3;5

rank n on m elements; U

n;m

aptures the idea of m points in general position in

rank n. The matroid U

3;5

is shown in Figure 23.

Note that U

3;5

does not have U

2;4

as a restri tion. Is U

3;5

binary? No. The

problem is that the matroid U

3;5

ontains U

2;4

as a proje tion or, in matroid termi-

nology, a ontra tion, as we will now see. (See Figure 24.) Consider the matroid

U

3;5

with ground set fA;B;C;D;Xg. If we ould realize U

3;5

as a restri tion of

a proje tive geometry, then the points of U

3;5

span a plane of this geometry, so it

suÆ es to view U

3;5

as a restri tion of PG(2; q). The lines `(A;X) and `(B;X)

of PG(2; q) must interse t the line `(C;D) of PG(2; q) sin e these are lines of a

proje tive plane. These points of interse tion, A

0

and B

0

, are distin t sin e A, X ,

and B are not ollinear, and similarly neither A

0

nor B

0

an be either C or D.

Therefore the line `(C;D) of PG(2; q) has at least four points, that is, PG(2; q) has

a restri tion isomorphi to U

2;4

. Therefore q annot be 2, so U

3;5

is not binary.

If we fo us on the line fA

0

; C;D;B

0

g in Figure 24 and relabel the points A

0

and

B

0

with the points A and B that proje ted onto these points, we get a geometry

in whi h the ats are pre isely that ats of U

3;5

that ontained X , but now with

X removed. The ats of U

3;5

that ontained X are fXg, fX;Ag, fX;Bg, fX;Cg,

fX;Dg, and fX;A;B;C;Dg; the ats of the proje tion are ;, fAg, fBg, fCg,

fDg, and fA;B;C;Dg. The generalization of this is the following very important

operation.


A

CDE

FX

A E

F C

B

B

D

E

B D

F C

Figure 25. The non-Fano matroid F

7

and its ontra tions by

fXg and by fX;Ag.

De�nition 6.2. Let M be a matroid on the set S and let Z be a subset of S. The

ontra tion M=Z of M by Z is the matroid on S � Z that has as its olle tion of

ats

fF � Z j F is a at of M with Z � Fg:

For more examples of ontra tion, Figure 25 shows the ontra tions of the non-

Fano matroid F

7

by fXg and by fX;Ag. Note that the ontra tion F

7

=fXg is not

a geometry: the non-singleton sets fA;Eg and fF;Cg are points. This shows that

applying a basi operation to a geometry does not ne essarily produ e a geometry;

to obtain a geometry, we would need to take the simpli� ation of the ontra tion,

but there are sometimes good reasons for not doing this.

Note that there are only two ats in the ontra tion F

7

=fX;Ag, namely fEg

as the least at and fE;B; F; C;Dg as the sole rank-1 at. The element E of the

rank-0 at of F

7

=fX;Ag is drawn as a hollow dot o� to the side sin e it does not

determine a point. The ontra tion F

7

=fX;Ag fails both onditions required of a

geometry | the empty set is not a at and the points are not singletons.

Note that F

7

=fXg=fAg= F

7

=fX;Ag; this is true in general for ontra tions by

a sequen e of disjoint sets. This and the orresponding result for deletions by a

sequen e of disjoint sets follow immediately from the de�nitions of these operations.

While it was immediate that restri tions of representable matroids are repre-

sentable, it is perhaps somewhat less transparent that the orresponding statement

about ontra tions is also true.

Theorem 6.3. If a matroidM is representable over GF (q), then every ontra tion

of M is representable over GF (q).

This result holds for arbitrary �elds, but, again, our hief interest is in �nite

�elds. One way of seeing Theorem 6.3 is to observe that ontra ting a subspa e of

a ve tor spa e is essentially taking the quotient of the ve tor spa e by the subspa e;

in the pro ess of taking this quotient, a representation of a matroid in the ve tor

spa e is arried to a representation of the orresponding ontra tion in the quotient

spa e.

The middle diagram in Figure 25 shows that the non-Fano matroid F

7

ontains

the 4-point line U

2;4

as a restri tion of a ontra tion. Therefore, by Theorem 6.3,

the non-Fano matroid F

7

is not binary. It is often useful to ombine restri tions

and ontra tions as illustrated in this example.

De�nition 6.4. A minor of a matroid M is any matroid that an be obtained from

M by repeatedly applying the operations of restri tion and ontra tion.

26 JOSEPH E. BONIN

It turns out that the operations of restri tion and ontra tion ommute, so every

minor of a matroid M amounts to a ontra tion of a restri tion of M .

From Theorems 6.1 and 6.3, we get Theorem 6.5.

Theorem 6.5. If a matroid M is representable over GF (q), then every minor of

M is representable over GF (q).

We an now state what has so far been the most su essful of the several paths

that have been pursued in the attempt to hara terize representability. Sin e mi-

nors of matroids that are representable over GF (q) are also representable over

GF (q), one ould hara terize representability by �nding the minor-minimal ma-

troids that are not representable over GF (q). That is, one will have hara terized

representability if one an �nd the minor-minimal obstru tions to representability.

The ex luded minors for representability over GF (q) are these the minor-minimal

obstru tions, the minor-minimal matroids that are not representable over GF (q).

The �rst ex luded minor hara terization of representability was proven by Tutte

in 1958. We have already seen that the four-point line is not binary; Tutte [32℄

showed that U

2;4

is the only ex luded minor for representability over GF (2).

Theorem 6.6. A matroid is binary if and only if it has no minor isomorphi to

U

2;4

.

There are a variety of proofs of Theorem 6.6; while most are algebrai , the proof

in [6℄ is geometri .

In [32℄, Tutte also gave the following hara terization of the matroids that are

representable over every �eld.

Theorem 6.7. A matroid is representable over every �eld if and only if it has no

minor isomorphi to any of U

2;4

, PG(2; 2), and a single-element deletion AG(3; 2)nx

of AG(3; 2).

AÆne and proje tive geometries have transitive automorphism groups, so there

is, up to isomorphism, only one single-element deletion AG(3; 2)nx of AG(3; 2).

The matroids that are representable over all �elds are alled regular matroids;

they are also known as unimodular matroids sin e they are the matroids that an

be represented by totally unimodular matri es (matri es with entries among the

integers 0, 1, or �1 in whi h all subdeterminants are 0, 1, or �1). Unimodular

matroids play an important role in optimization.

Sin e lines in PG(n � 1; q) have q + 1 points, the (q + 2)-point line U

2;q+2

is

an ex luded minor for representability over GF (q), but for q > 2, there will be

additional ex luded minors. Theorem 6.8, the hara terization of ternary matroids,

or matroids that are representable over GF (3), illustrates this. Ternary matroids

were hara terized in 1971 in unpublished work of R. Reid. The �rst published

proofs were by Bixby [4℄ and Seymour [30℄ in 1979. Seymour's proof introdu ed

many new ideas that revolutionized work on representability and other aspe ts of

matroid theory.

Theorem 6.8. A matroid is ternary if and only if it has no minor isomorphi to

U

2;5

, U

3;5

, PG(2; 2), or AG(3; 2)nx.

As a orollary of Theorems 6.6, 6.7, and 6.8, we have the following theorem.

Theorem 6.9. A matroid is representable over every �eld if and only if it is

representable over both GF (2) and GF (3).

Re ent work of Whittle [37℄ studies matroids that are representable over other

pairs of �elds of relatively prime hara teristi .


The ex luded minors for matroids that are representable over GF (4) have been

found re ently by Geelen, Gerards, and Kapoor [20℄. There are seven ex luded

minors, the largest two of whi h have eight points and rank four.

One of the outstanding problems in this area is the following onje ture due to

Gian-Carlo Rota.

Conje ture 6.10. For ea h prime power q, the number of ex luded minors for

representability over GF (q) is �nite.

Rota made this onje ture in 1971, when there was barely any eviden e for it. His

motivation for the onje ture ame from invariant theory. Re ent work of Johnson,

Seymour, and Robertson suggests a strategy to prove Rota's onje ture that has

grown out of the graph minors proje t; this will be an ex iting area to wat h.

Re ently Oxley, Semple, and Vertigan [28℄ gave the following exponential lower

bound on the number of ex luded minors for representability over GF (q).

Theorem 6.11. There are at least 2

q�4

ex luded minors for representability over

GF (q).

Note that 2

q�4

is well below the number of ex luded minors mentioned above

for GF (2), GF (3), and GF (4). Looking at the strategy in [28℄ reveals that their

argument overs only ex luded minors of a relatively simple type; the ex luded mi-

nors they ount are all variations on the (q+2)-point line obtained by the ex hange

operations they develop. This suggests that the a tual number of ex luded minors

is extremely large.

We have seen that, apart from the ase q = 2, the (q + 2)-point line U

2;q+2

is not the only minor one has to ex lude in order to hara terize representability

over GF (q). However, as the following theorem from [6℄ shows, if there are enough

points relative to the rank, then ex luding only U

2;q+2

as a minor is enough to

guarantee representability.

Theorem 6.12. Assume that q is a prime power and n ex eeds three. Any rank-n

geometry with no U

2;q+2

minor and with at least q

n�1

points is representable over

GF (q).

Assume q is an odd prime power and n ex eeds three. Any rank-n geometry with

no U

2;q+2

minor and with at least q

n�1

� (q

n�2

� 1)=(q� 1) points is representable

over GF (q).

This result leads to the next major bran h of matroid theory we will glimpse,

extremal matroid theory. Theorem 6.12 and related results in [6℄ strengthen the

result of Kung [25℄ that opens the next se tion.

7. Extremal Matroid Theory

Theorem 6.12 assumed that q is a prime power be ause �nite �elds have prime

power order. However, it is possible to onsider the e�e t of ex luding the (q + 2)-

point line minor even if q is not a prime power. A basi result along these lines is

the following theorem of Kung [25℄.

Theorem 7.1. Rank-n geometries that have no (q + 2)-point line minor have

at most (q

n

� 1)=(q � 1) points. This upper bound is attained only by proje tive

geometries of order q.

Thus, this upper bound is stri t if there is no proje tive geometry of rank n and

order q (e.g., if q is not a prime power and n is at least 4).

28 JOSEPH E. BONIN

Theorem 7.1 aptures the spirit of extremal matroid theory: a ondition (ex lud-

ing U

2;q+2

as a minor) is imposed on matroids and the e�e t on a parameter (the

number of points) is analyzed. Another result of this type is the following theorem

from [6℄ that gives a ounterpart of Theorem 7.1 for aÆne geometries.

Theorem 7.2. Rank-n geometries that have no U

2;q+2

minor and have no (q+1)-

point lines have at most q

n�1

points. This upper bound is attained only by aÆne

geometries of order q.

It is natural to ask if the upper bound in Theorem 7.1 an be improved signi�-

antly when q is not a prime power. In [6℄ and [8℄, a number of su h improvements

are given, but those upper bounds are not sharp. The only urrently known sharp

upper bound for a problem of this type is the following theorem from [8℄. This

addresses the smallest integer that is not a prime power, q = 6, so q + 2 is 8.

Theorem 7.3. Let n be greater than 3 and let M be a rank-n geometry with no

8-point line minor. Then the number of points in M is at most (5

n

� 1)=(5 � 1).

This upper bound is sharp and is attained only by the rank-n proje tive geometry

PG(n� 1; 5).

Theorem 7.3 proves the �rst ase of the following onje ture of Kung [25℄.

Conje ture 7.4. Let q

�

be the largest prime power not ex eeding q. For all suÆ-

iently large n, the greatest number of points a rank-n geometry that has no U

2;q+2

minor is (q

n

�

� 1)=(q

�

� 1). This upper bound is sharp and is attained only by the

rank-n proje tive geometry PG(n� 1; q

�

).

This onje ture, whi h appears to be very diÆ ult, asserts that proje tive ge-

ometries play an extremely entral role in problems that have U

2;q+2

as an ex luded

minor. It would be a solid advan e to prove Conje ture 7.4 even for q = 10.

To relate these new topi s to our motivating examples, we an now hara terize

lassi al geometries through extremal matroid theoreti properties. Theorems 7.1

and 7.2 are results of this sort, and they have as further orollaries a number

of additional hara terizations of proje tive and aÆne spa es; the following two

theorems, from [9℄ and [13℄ respe tively, sample a few su h results.

Theorem 7.5. A rank-n geometry with (q

n

� 1)=(q � 1) points in whi h all lines

have at least q + 1 points is a proje tive geometry of order q.

A rank-n geometry with q

n�1

points in whi h all lines have q points and all planes

have at least q

2

points is an aÆne geometry of order q.

Assume q > 2. A rank-n geometry with q

n�1

points in whi h all lines have q

points and all hyperplanes have q

n�2

points is an aÆne geometry of order q.

Theorem 7.6. Let q > 1 be an integer. Let M be a rank-n geometry in whi h ea h

hyperplane ontains all but at most q

n�1

elements of M and in whi h all lines have

at least q + 1 points. Then the number of points in M is exa tly (q

n

� 1)=(q � 1)

and M is a proje tive geometry of order q.

Theorem 7.1 suggests other interesting questions. By ex luding U

2;q+2

as a

minor, one gets an upper bound of (q

n

� 1)=(q � 1) on the number of points. It

follows that there are upper bounds on the number of lines, the number of planes,

and, in general, the number of rank-i ats, but what are the optimal upper bounds?

Open Problem 7.7. Assume that q is a prime power and, if needed, n is suÆ-

iently large. Is the number of rank-i ats of PG(n� 1; q) an upper bound on the

number of rank-i ats in any rank-n geometry that has no U

2;q+2

minor? If so, are

proje tive geometries the only geometries that attain these bounds?


In some extremal problems, one pla es restri tions on the subgeometries of a

geometry, rather than on all minors. An instan e of this is the following theorem,

whi h is a orollary of Theorem 7.2.

Theorem 7.8. Rank-n geometries that are representable over GF (q) and have no

lines with q + 1 points have at most q

n�1

points. Furthermore, AG(n� 1; q) is the

only su h geometry with q

n�1

points.

Note that the (q + 1)-point line is PG(1; q); note also that AG(n � 1; q) is

PG(n � 1; q)nPG(n � 2; q). It would not be surprising if other deletions of the

form PG(n � 1; q)nPG(k � 1; q) arose from ex luding other proje tive geometries

as subgeometries. The following theorem of Bose and Burton [14℄ treats this.

Theorem 7.9. Subgeometries of PG(n�1; q) that have no subgeometries isomor-

phi to PG(m� 1; q) have at most

q

n

� q

n�m+1

q � 1

points. Furthermore, PG(n � 1; q)nPG(n � m; q) is the only su h geometry with

(q

n

� q

n�m+1

)=(q � 1) points.

Note that Theorem 6.12 allows us to repla e the hypothesis of representability

over GF (q) in Theorem 7.9 with the weaker hypothesis of no minors isomorphi to

U

2;q+2

. Thus, rank-n geometries that have no minors isomorphi to U

2;q+2

and no

subgeometries isomorphi to PG(m�1; q) have at most (q

n

�q

n�m+1

)=(q�1) points,

with the only example that attains this bound being PG(n� 1; q)nPG(n�m; q).

To end this brief look at extremal matroid theory, we mention a theorem and a

onje ture of a very di�erent avor. We �rst establish some notation for dealing

with the types of problem en ountered in Theorem 7.9. Let ex

q

(M ;n) be the

maximum number of points in a rank-n geometry that is representable over GF (q)

and that has no restri tion isomorphi to the geometry M . Thus, Theorem 7.9

determines ex

q

(PG(m� 1; q);n). The following theorem appears in [11℄.

Theorem 7.10. Assume that M a subgeometry of AG(m� 1; 2). Then

lim

n!1

ex

2

(M ;n)

2

n

� 1

= 0:

In other words, for any subgeometry M of AG(m � 1; 2), the \size fun tion"

ex

2

(M ;n) is an order of magnitude smaller than 2

n

� 1, the number of points

in PG(n � 1; 2). This is the strongest bit of eviden e to date for the following

onje ture, whi h, although never stated, appears between the lines in [25℄.

Conje ture 7.11. If the geometry M is representable over GF (q), then

lim

n!1

ex

q

(M ;n)

(q

n

� 1)=(q � 1)

= 1� q

� +1

where is the minimum number of subgeometries of AG(m � 1; q) into whi h M

an be partitioned.

In general, it is extremely diÆ ult to ompute ex

q

(M ;n). (See [11℄ for all ur-

rently known results on ex

q

(M ;n).) Conje ture 7.11 is an attempt to des ribe the

asymptoti behavior of ex

q

(M ;n).

30 JOSEPH E. BONIN

M M1 2

Figure 26. Two matroids with the same Tutte polynomial.

8. Matroid Invariants

The �nal bran h of matroid theory we will brie y dis uss is the theory of Tutte

polynomials and other invariants of matroids. The Tutte polynomial is a two-

variable polynomial asso iated with a matroid. For a matroidM on the ground set

S, the Tutte polynomial t(M ;x; y) is de�ned as follows:

t(M ;x; y) =

X

A�S

(x� 1)

r(S)�r(A)

(y � 1)

jAj�r(A)

:(2)

This polynomial generalizes many important polynomials in mathemati s, in luding

the hromati and ow polynomials of graph theory, weight enumerators in oding

theory, the Jones polynomial for alternating knots, and, in some ases, the partition

fun tion of the Ising model in statisti al physi s.

The Tutte polynomial ontains mu h information about the matroid; for in-

stan e, from the Tutte polynomial, one an determine the rank of the matroid,

the ardinality of the ground set, the ardinalities of suÆ iently large ats of ea h

rank, and many more parameters that we have not dis ussed. Still, nonisomorphi

matroids an have the same Tutte polynomial. This is illustrated by the matroids

in Figure 26. For these matroids, we have

t(M

1

;x; y) = t(M

2

;x; y) = (x� 1)

3

the empty set

+6(x� 1)

2

the six singleton sets

+15(x� 1) the �fteen pairs

+18 eighteen of the 3-subsets have rank 3

+2(x� 1)(y � 1) the other two have rank 2

+15(y � 1) the 4-element subsets

+6(y � 1)

2

the 5-element subsets

+ (y � 1)

3

the entire set.

There are several general te hniques for produ ing more examples of matroids

with the same Tutte polynomial (see, e.g., [7, 12, 16℄). Before turning to some

omplementary results (matroids that are determined by their Tutte polynomial),

we sket h some of the reasons that Tutte polynomials are so important.

A number of frequently-studied invariants satisfy \deletion- ontra tion" rules.

One of the oldest invariants of this type is the hromati polynomial of a graph.


The hromati polynomial, �(G;x), of a graph G assigns to ea h positive integer x

the number of proper olorings of the verti es of G with x \ olors" (whi h an be

taken to be the integers 1; 2; : : : ; x); proper means that verti es that are joined by

an edge must be assigned di�erent olors. Noti e that for an edge e of a graph G,

we have

�(G� e;x) = �(G;x) + �(G=e; )(3)

where G�e is the deletion, G with the edge e removed, and G=e is the ontra tion,

the graph formed from G by removing e and identifying the verti es that e joined.

(These operations are intimately linked with the like-named matroid operations.)

Equation (3) holds sin e any proper oloring of G� e either assigns the endpoints

of e di�erent olors (giving a proper oloring of G) or the same olors (giving a

proper oloring of G=e). It is useful to write Equation (3) in the form

�(G;x) = �(G� e;x)� �(G=e; ):(4)

This is a re ursive expression for �(G;x). From this, one an indu t to show that

�(G;x) is indeed a polynomial fun tion of x.

One an show that the Tutte polynomial also satis�es a deletion- ontra tion rule.

Indeed, an alternative way of de�ning the Tutte polynomial is as follows. The Tutte

polynomial of matroid whose ground set is the empty set is 1. If e is an element of

a matroid M , then

t(M ;x; y) =

8

>

<

>

:

t(Mnfeg;x; y) + t(M=feg;x; y) if r(feg) > 0 and r(Mnfeg) = r(M);

y t(M=feg;x; y) if r(feg) = 0;

x t(Mnfeg;x; y) if r(Mnfeg) < r(M):

From this formulation, it is not immediately lear that the Tutte polynomial is

well-de�ned; on eivably it ould depend on the order in whi h the elements of M

are deleted and ontra ted. However, sin e this is equivalent to Equation (2), the

Tutte polynomial is indeed well-de�ned.

The Tutte polynomial is important in part be ause it satis�es the deletion-

ontra tion rule but more so be ause it is the universal invariant for all invariants

that satisfy a deletion- ontra tion rule; all other invariants of matroids that satisfy

a deletion- ontra tion rule are evaluations of the Tutte polynomial. The following

theorem makes this pre ise.

Theorem 8.1. Let R be a ommutative ring that has unity. For ea h hoi e of

elements u; v; �; � of R, there is a unique fun tion T from the lass of all matroids

into R that has these properties.

(i) If M is the matroid on the empty set, then T (M) = 1.

(ii) If e is an element of M and r(feg) > 0 and r(Mnfeg) = r(M), then

T (M) = � T (Mnfeg) + � T (M=feg):

(iii) If e is an element of M and r(feg) = 0, then T (M) = v T (M=feg).

(iv) If e is an element of M and r(Mnfeg) < r(M), then T (M) = uT (Mnfeg).

Furthermore, T is the following evaluation of the Tutte polynomial t(M ;x; y): for

a matroid M with a k-element ground set, we have

T (M) = �

k�r(M)

�

r(M)

t(M ;u=�; v=�):

32 JOSEPH E. BONIN

Brylawski [15℄ proved this in the ase of � = � = 1; Oxley and Welsh [29℄

observed that the same argument yields the general ase. That this result is very

useful makes it all the more striking that the proof is a simple indu tion based on

the deletion- ontra tion rule.

To sket h (very in ompletely) one relevant appli ation of Theorem 8.1, note

that in Equation (4) we have � = 1 and � = �1. The hromati polynomial of

a graph that onsists of a single edge is x(x � 1) sin e one vertex an be olored

with any of x olors and the other vertex an be olored with any of the x � 1

other olors. A graph with an edge that is in ident with only one vertex (a loop)

has no proper olorings. With a graph G, there is an asso iated matroid, the

y le matroid M(G). Skipping many steps, one an argue from this that �(G;x) is

x

!(G)

(�1)

r(M(G))

t(M(G); (x � 1)=(�1); 0=1), or

x

!(G)

(�1)

r(M(G))

t(M(G); 1� x; 0)(5)

where !(G) is the number of omponents of the graph G.

Before pursuing su h spe ializations of the Tutte polynomial further, we mention

more results about Tutte polynomials. The example at the beginning of this se tion

shows that several matroids an have the same Tutte polynomial; we mentioned that

there are general onstru tions for produ ing more su h examples. At the opposite

end of the spe trum, some matroids are determined by their Tutte polynomials.

The next theorem is from [9℄.

Theorem 8.2. If n � 4 and t(M ;x; y) = t(AG(n� 1; q);x; y), then M is isomor-

phi to AG(n� 1; q). If t(M ;x; y) = t(AG(2; q);x; y), then M is an aÆne plane of

order q.

There are extremely few results of this type known. The orresponding result

for PG(n�1; q) was known earlier (see [16℄) and has many simple proofs. To prove

Theorem 8.2, one an show that the hypotheses of the �rst hara terization of aÆne

geometries in Theorem 7.5 an be veri�ed from the Tutte polynomial of a matroid.

Uniform matroids are also determined by their Tutte polynomials, as are the y le

matroids of omplete graphs and ertain generalizations of su h matroids (see [9℄).

Re all that AG(n � 1; q) is PG(n � 1; q)nPG(n � 2; q). With this in mind, we

see that the following result from [1℄ in a natural follow-up to Theorem 8.2.

Theorem 8.3. Assume n and k are integers with n � 3 and 1 � k � n � 2.

Assume the matroid M has the same Tutte polynomial as PG(n�1; q)nPG(k�1; q).

If n � 4, then M is isomorphi to PG(n� 1; q)nPG(k � 1; q). If n = 3, then M is

isomorphi to some single-element deletion of some proje tive plane of order q.

One key element of the proof of Theorem 8.3 and related results in [1℄ is that

the geometries PG(n�1; q)nPG(k�1; q) are the unique geometries that attain the

bound in a result of extremal matroid theory, spe i� ally, Theorem 7.9.

In ontrast to su h results, huge olle tions of representable matroids an have

the same Tutte polynomial. In [7℄, te hniques are developed to produ e many

families of su h olle tions; the following theorem is the tip of the i eberg in this

area.

Theorem 8.4. Assume that q is the t-th power of a prime, that q ex eeds 5, and

that q�1 is not a Mersenne prime. Let d be the largest proper divisor of q�1. For

ea h integer n with n � 3, there are at least

(d� 1)

n�1

(q � 1� d)(q � 2� d)

t


non-isomorphi 3- onne ted geometries that are representable over GF (q), that have

rank 2n+ 2, that ontain

q

2n+2

� 1

q � 1

� 5n� 5

points, and that have the same Tutte polynomial. In parti ular, if q is odd, then

there are at least

�

q � 3

2

�

n

q � 1

2t

su h geometries.

We mention two open problems along these lines.

Open Problem 8.5. Find more matroids that are uniquely determined by their

Tutte polynomials.

Open Problem 8.6. Find more onstru tions that produ e nonisomorphi ma-

troids that have the same Tutte polynomial.

The parti ular evaluation of the Tutte polynomial in Equation (5) arises so

frequently that it is singled out. The hara teristi polynomial of a matroid M is

de�ned by

p(M ;x) = (�1)

r(M)

t(M ; 1� x; 0):

Thus, up to a power of x, the hara teristi polynomial p(M ;x) is a generalization

of the hromati polynomial of a graph.

The hromati polynomial of a graph has the following simple property: the �rst

so many positive integers are roots of the hromati polynomial and then there is

never another positive integer root of this polynomial. This is simply be ause if

G annot be olored with k olors, then G annot be olored with k � 1 olors.

One an show that matroids that are representable over GF (q) have an analogous

property with respe t to powers of q; the �rst so many powers of q are roots of the

hara teristi polynomial and then there are no more powers of q that are roots

of this polynomial. Thus, in analogy with the hromati number of a graph, the

smallest integer k whi h is not a root of the hromati number (equivalently, the

smallest integer k su h that the graph an be olored with k olors), we de�ne the

riti al exponent of a geometry that is representable over GF (q). (We ould de�ne

the riti al exponent of a matroid in general but the riti al exponent depends only

on the simpli� ation of the matroid. Therefore the theory of riti al exponents

generally fo uses on geometries.)

De�nition 8.7. Assume that M is a geometry that is representable over GF (q).

The riti al exponent of M over GF (q) is the least positive integer k su h that

p(M ; q

k

) 6= 0.

The following theorem of Crapo and Rota [18℄ begins to hint at the importan e

of the riti al exponent.

Theorem 8.8. Assume that the rank-m geometry M is representable over GF (q)

and has riti al exponent over GF (q). For any n � m and any embedding of M

in PG(n� 1; q), the least odimension of a subspa e of PG(n� 1; q) disjoint from

M is .

It is striking that this odimension does not depend on the embedding or even

the dimension in whi h the matroid is embedded! One an argue from this that the

34 JOSEPH E. BONIN

riti al exponent is also the smallest number of subgeometries of AG(n� 1; q) into

whi hM an be partitioned. Thus, the in Conje ture 7.11 is the riti al exponent

of M .

We are approa hing the riti al problem of matroid theory, a huge area of matroid

theory that onne ts with many important topi s. For instan e, if we understood

riti al exponents well enough, we would have a solution to the fundamental prob-

lem of linear oding theory [19℄. While it would be fas inating to pursue this topi

here, we refer the reader to Kung [26℄ for a state-of-the-art survey.

In Theorems 8.2 and 8.3, and the related results mentioned above, we saw that

some matroids are uniquely determined by their Tutte polynomials. We lose this

se tion with a few analogous results about hara teristi polynomials from [1℄.

Theorem 8.9. Assume that the geometry M is representable over GF (q) and that

p(M ;x) = p(PG(n�1; q)nPG(n�m; q);x). Then M is isomorphi to the geometry

PG(n� 1; q)nPG(n�m; q).

Theorem 8.10. Assume n � 4. If p(M ;x) = p(PG(n � 1; q)nPG(n � m; q);x)

and the multiset

fp(M=X ;x) j X a at of M of rank n� 2g

is equal to the orresponding multiset onstru ted from PG(n�1; q)nPG(n�m; q),

then M is isomorphi to PG(n� 1; q)nPG(n�m; q).

The proof of Theorem 8.9 rests on the observation, following from Theorem 8.8,

that PG(n � 1; q)nPG(n � m; q) is the unique largest geometry of rank n that

has riti al exponent n � m + 1. Theorem 8.10 uses this observation along with

Theorem 6.12.

9. Con lusion

This introdu tion to matroid theory has s rat hed the surfa e of only a few

of the many areas to be explored in the subje t. We have not at all addressed

maps between matroids (strong maps, weak maps, omaps) [33℄, onstru tions [27,

33℄, basis ex hange properties [33℄, stru ture theory and onne tivity [10, 27℄, the

splitter theorem and its onsequen es [10, 27℄, enumeration [18, 31℄, and the entire

�eld of oriented matroids [5℄. We have also omitted the numerous appli ations,

whi h in lude optimization [34℄, oding theory [17, 19, 26, 31, 34℄, onne tions with

di�erential geometry, stru tural rigidity [10, 21, 35℄, arrangements of hyperplanes,

and mu h more. It is hoped that this introdu tion whets your appetite to pursue

matroids further.

Referen es

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Department of Mathemati s, The George Washington University, Washington, DC

20052

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cpb-us-e1.wpmucdn.com · A BRIEF ODUCTION INTR TO OID TR MA Y THEOR JOSEPH E. BONIN 1. oduction Intr y Man t di eren topics in mathematics, ranging from applied sub jects h suc as