The Cofinality of the Random Graph

The Cofinality of the Random GraphAuthor(s): Steve WarnerSource: The Journal of Symbolic Logic, Vol. 66, No. 3 (Sep., 2001), pp. 1439-1446Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2695116 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 66, Number 3, Sept. 2001

THE COFINALITY OF THE RANDOM GRAPH

STEVE WARNER

Abstract. We show that under Martin's Axiom, the cofinality cf (Aut(F)) of the automorphism group of the random graph F is 2'.

?1. Introduction. Suppose that G is a group which is not finitely generated. Then G can be expressed as the union of a chain of proper subgroups. The cofinality of G, written cf (G), is defined to be the least cardinal A such that G can be expressed as the union of a chain of A proper subgroups. In [6], Macpherson and Neumann proved that cf (Sym(co)) > co. In [7], Sharp and Thomas proved that it is consistent that cf (Sym(co)) and 2a can be any two prescribed regular, uncountable cardinals, subject only to the obvious requirement that cf (Sym(co)) < 2a. Then, in [8], Sharp and Thomas considered the relationship between cf (Sym(co)) and two well-known cardinal invariants of the continuum, the dominating number D and the bounding number b. They proved that cf (Sym(co)) < 0, and that both cf (Sym(co)) < b and b < cf (Sym(co)) are consistent with ZFC .

If we regard Sym(co) as the automorphism group of (co; ), the "trivial countably infinite structure," then it is natural to try to compare cf (Sym(co)) and cf (Aut(X)), where 4 is a countable structure. In [9], Thomas showed that if X is co-categorical, then cf (Aut(ld)) < cf (Sym(co)). There exist countable co-categorical structures X such that cf (Aut(Qt)) < cf (Sym(co)). For example, in [5], Lascar showed that there exists a countable co-categorical structure q such that the product of countably many cyclic groups of order 2 is a homomorphic image of Aut(Q). It follows that cf (Aut(?W)) = co. In [9], Thomas also showed that if X is a vector space over a finite field F, then cf(Aut(/t)) = cf(Sym(co)). On the other hand, the following question is open:

Question. Is it consistent that there exists a countable co-categorical structure 4 such that

co < cf (Aut(ld)) < cf (Sym(wo))?

We denote by F the random graph (see [1, pp. 37-38 ]) which is uniquely character- ized up to isomorphism among graphs on countably many vertices by the following property:

*) If U. V are disjoint, finite sets of vertices in F, then there is a vertex x of F which is adjacent to all vertices in U and to no vertices in V.

Received February 11, 2000; revised July 18, 2000.

? 2001, Association for Symbolic Logic

0022-4812/01/6603-0027/$1 .80

1439



1440 STEVE WARNER

In [2], Hodges, Hodkinson, Lascar, and Shelah showed that cf (Aut(F)) > C. In this paper, it will be shown that under Martin's Axiom, cf (Aut(F)) = 2 .

DEFINITION 1. 1. (1) MA (i) is the statement that if F is a nonempty c..c. partial order, and 9 a family of dense subsets of F with IO I < A, then there exists a filter , C F such that ' n D 74 0 for every D e 9.

(2) Martin's Axiom (MA) is the statement that MA (A) holds for all A < 2w.

The main result to be proved is the following: THEOREM 1.2. Let F be the random graph and let G = Aut(F). If A is a regular

cardinal, then MA(A) - cf (G) > A. From now on G = Aut(F). We will prove Theorem 1.2 in the following way.

Suppose that G = U,<H0,. We will use MA (A) to construct a "generic" sequence of automorphisms (g0,),<A (as defined in [2] and [10]) and a strictly increasing sequence (JA< such that go, e Hza \ Up<EaHp for each a < A. We will then use MA(A) again to find an element vP e G such that for each a < A, there is a fi > a such that TgT-1 = gp. Then for some a < A, T E Ha C Hip. So gp = Tgs, T-1 E ,e , a contradiction. All of this will be made precise later in this paper.

Theorem 1.2 and the proof of Corollary 2.2 in [7] give us the following result: COROLLARY 1.3. Let M F GCH and suppose that A < 0 are regular uncountable

cardinals in M. Then there exists a c.c.c. poset F such that

Mp F cf(G) = < 0 =2.

The results of this paper form part of the author's Ph.D. thesis written under the supervision of Simon Thomas. The author would like to thank Simon Thomas for his encouragement and helpful discussions.

?2. Generic sequences of automorphisms. In this section, following Truss [10], we define the notion of a generic sequence of automorphisms. The existence of generic sequences of countable length was proved in [2]. Since many of the ideas in [2] are central to the proof of Theorem 1.2, we repeat the relevant results in this section. However, we will rewrite this exposition in the language of games, as we feel that it is slightly easier to understand in this form. We then show that under MA (A), we can construct generic sequences of length A.

If A C 1, then we denote by GA the pointwise stabilizer of A. It is well known that G is a Polish group with basis

{gGF geG, FCF, IFI <wc}.

The open subgroups of G are precisely the subgroups which contain the pointwise stabilizer of a finite set.

LEMMA 2.1. If H is an open subgroup of G, then there are only finitely many subgroups K of G that contain H.

PROOF. See [2, Lemma 2.4]. -

In particular, if G = Ua<AH,,H, then Ha is not open for each a < A.

DEFINITION 2.2. (1) If g, h e G, then the conjugate of g by h is defined to be gh = hgh'1.



THE COFINALITY OF THE RANDOM GRAPH 1441

(2) If (gi. g,)e Gn, then the corresponding conjugacy class is defined to be

(gi g,') (gh X.. ) heG}

Recall that a subset C C G is comeagre if C contains a countable intersection of dense open subsets of G. A subset M C G is meagre if G \ M is comeagre.

DEFINITION 2.3. (g1, . . . gn) e Gn is said to be generic if (gi . . .g, )G is comeagre in Gn (in the product topology).

Note that there can be at most one generic conjugacy class, for if (gI, gn )G

and (hl,... hn )G are both comeagre, then they must intersect. But conjugacy classes are either disjoint or coincide. So (gl, ... , g ) G = (hi, , hn ) G.

Notation (F,rgs... gn) is the obvious expansion of F to a language with n new 1-place function symbols.

PROPOSITION2.4. Let (gi,... ,gn), (h. 1 hn) e Gn. Then

(hi,.. hn) e(g , *1, gn)G iff(F, gi, gn) (F, , han)

In particular, if (gI. gn), (hi,... , hn) are both generic sequences, then

(F, gi. , gn) (jF, h,. hn).

PROOF. (F, g, g n)g (F, hi,... , hn) iffthere is a k e G such that k(gi (a))- hi(k(a)) for all a e F and i = 1, . . ., n if there exists k e G such that hi = gfC for alli = 1,... ,n. -

We will use Banach-Mazur games to show that there is a generic conjugacy class. For each A C G let b (A) be the game defined as follows. Let

P = {f f: - F I f is a finite partial isomorphism}

where F is ordered by p < q iff p q. Then in the game 05(A), Players I and II choose a decreasing sequence

PO>P > >..._Pn >?... ,neco

of elements of P. Player I chooses Pi iff i is even. Player II wins if UnEpn e A.

Remark. Player II can easily ensure that UnEco, Pn e G.

THEOREM 2.5. Player II has a winning strategy in 0!(A) iff A is comeagre in G.

PROOF. See [4, Theorem 8.33]. A

THEOREM 2.6. Player I has a winning strategy in 6!(A) if there is a finite F C F and g e G such that A n gGF is meagre in gGF.

PROOF. See [4, Theorem 8.33]. -

COROLLARY 2.7. g e G is generic iff Player II has a winning strategy in 05(g G).

By the next result, if H is not open, then Player I does not have a winning strategy in ! (G \ H). This will allow us to construct a generic automorphism which is not in H.

THEOREM 2.8. Let H be a subgroup of G. Then H is open if Player I has a winning strategy in !5(G \ H).



1442 STEVE WARNER

PROOF. If H is open, then there is a finite F C F such that GF < H. As his first move, let Player 1 play PO = id [ F. Then clearly Player I can ensure that UnECoPn e GF.

Conversely, suppose that Player I has a winning strategy in hi (G \H). By Theorem 2.6, there is a finite F C F and g e G such that (G \ H) n gGF is meagre in gGF-

Then H n gGF is comeagre in gGF. In particular, we have that H n gGF 7E O. So, we may assume that g e H. Therefore, H n gGF = g(H n GE) It follows that H n GF is comeagre in GF, and hence the same is true of each coset of H n GF in GF. Since distinct cosets are disjoint, we have that GF = H n GF. Thus GF < H, and so H is open. A

We now show that a generic conjugacy class exists. Consider the class v of all structures of the form (X, f) where X is a finite graph, and f e Aut(X). We say that ( Y. g) is a substructure of (X, f ) iff Y C X and g = f [ Y. Clearly v has the amalgamation property, and hence there exists a unique countable universal homogeneous structure (A, (p) with respect to X.

CLAIM 2.9. A is the random graph.

To prove this claim, we will need the following theorem of Hrushovski.

THEOREM 2.10 (Hrushovski). Let X be a finite graph. Then there exists a finite graph Z, containing X as an induced subgraph, such that any isomorphism between induced subgraphs of X extends to an automorphism of Z.

PROOF. See [3]. -

PROOF OF CLAIM 2.9. Suppose that (X, f) e v and that X = UU V is a partition. Let Y = X U {x}, where x is a new vertex which is adjacent to all the vertices in U and to none in V. By Hrushovski's Theorem, there exists a finite graph Z D Y such that any isomorphism between subgraphs of Y extends to an automorphism of Z. In particular, f extends to an automorphism g of Z and (Z, g) e X. Claim 2.9 follows easily. A

THEOREM 2.1 1. p is a generic element of G.

PROOF. Consider a play of the game f5 (pG ), say

PO > PI >_.p > n >_ ... , n e co,

and let f = UnEcn - It is clear that Player II can play so that the following hold:

(1) f e G. (2) for all odd i, dom pi = ran pi. (Use Hrushovski's Theorem). (3) (F, f ) is existentially closed in the class of locally-XW structures.

It follows that (F, f ) is a universal homogeneous structure with respect to X. Thus (F, f ) (F, p). So by Proposition 2.4, f e FoG. -A

We can easily generalize the above to generic sequences (g1. gn) e Gn for each 1 < n < co. Let Ipn = xP X . IP. Then for each subset A C Gn, we have the obvious game ! (A).

THEOREM 2.12. Player II has a winning strategy in '5(A) ifeA is comeagre in Gn (in the product topology).




THEOREM 2.13. Player I has a winning strategy in 63(A) if there arefinite F1, ... Fn C F and gl,... ,gn e G such that A n (gi GF1 X .. X gnGF) is meagre in

g, GF X .X gn GFn.

COROLLARY 2.14. (g1, gn) e Gn is generic if Player II has a winning strategy in e(1(gi, . , gn )G).

Consider the class An of all structures of the form (X, f1,. , fn) where X is a finite graph, and f i e Aut(X) for each i = 1, . . ., n. As before, we have the obvious notion of a substructure, and since An has the amalgamation property, there exists a unique countable universal homogeneous structure (A, Sol... ., On) with respect to n. Using the same arguments as before, we see that A is the random graph, and that (Soj,... , (.n) is generic. We therefore have the following:

THEOREM 2.15. There exist generic (gi, . . . , gn) e G nfor each 1 < n < co.

THEOREM 2.16. If (g1, . . . , gn) is generic, then

A = {g e G I (g1 . gn, g) is generic}

is comeagre in G.

PROOF. See [4, Theorem 8.33]. -

To prove Theorem 1.2, we will need longer generic sequences.

DEFINITION 2.17. If ,B is any ordinal, then we say that (g)Q,<p is generic if for any n second a1 <.. <an </3, (gaQ1,., gan) e Gn isgeneric.

THEOREM 2.18 (MA (X)). Let ,B < A and (ga) a<p be generic. Then,

{gp e G I (ga)a<p isgeneric}

is comeagre in G.

PROOF. {gp e G I (ga)a<p is generic} is the intersection of the sets

{ f EEG I(gap ... ,gan, f )is generic} (n EEco,al < ... < an </

This is an intersection of < A comeagre sets and hence is comeagre by MA (A). -1

The next lemma will enable us to construct our desired generic sequence as stated in the Introduction.

LEMMA 2.19. Let C be comeagre in G, H a subgroup of G which is not open, and o: F -, F afinite partial isomorphism. Then there is a g e C \ H which extends So.

PROOF. Consider a play of the game Oi(C \ H), say

Po> P > ... Pn > ... ,n eco.

Let Player I use the following "strategy": On his first move Player I plays po = So. For the rest of the game, Player I pretends to be Player II using a winning strategy in the game 5 (C); and thus Player I will ensure that g = Un<,pPn E C. By Theorem 2.8, this strategy is not winning for Player I in 5 (G \ H). Hence Player II can ensure that g = Un<wpn E G \ H. -1



1444 STEVE WARNER

?3. Proof of Theorem 1.2. We are now ready to prove the main result of this paper.

THEOREM 1.2. Let F be the random graph and let G = Aut(F). If A is a regular cardinal, then MA(A) F cf (G) > A.

PROOF. Let K < A be a regular cardinal, and suppose that G = Us<,<H is a chain of proper subgroups. Then by [2], K > co. By Lemma 2.1, each Hc, is not open. We define a sequence (g,),<,< of elements of G and a strictly increasing sequence (WX)a<n of ordinals so that:

(i) For each a < X, ga E Hz \ Ufl<Ha.Hp (ii) For each a < X, (gp)p<a is generic.

(iii) Suppose that y < a < n and that p F -> F is a finite partial isomorphism such that dom p is invariant under gy. Then there exists /1 > a such that

(ogy ( -'(a) = gp (a) for all a e ran po.

As the set of generic elements of G is comeagre in G, we can choose a generic element go e G. Let 4o be the least ordinal such that go e H0.o Then go, do satisfy (i) and (ii).

Now suppose that (gp)p<a, has been defined satisfying (i) and (ii). Let J be the set of all pairs (y, p) satisfying the following conditions:

(a) y < a. (b) p : F -> F is a finite partial isomorphism. (c) dom p is invariant under gy.

Then J has cardinality ,u < a. Enumerate this set as ((QY, ())a<f<a+p, and for

each ,6, let hp - ~opgy fl~1 ran bp. We extend each ha+z to ga+z satisfying (i) and

(ii) as follows: Suppose (gp)p<?+,5 and (4P)p<a+ satisfy (i) and (ii). The set

C = {ga e G I (gp)p<a+i is generic}

is comeagre in G. Let p = sup{fp I ,6 < a + 6 }. By Lemma 2.19 there is

ga+,e C \ Hp such that ga+, extends ha+5. Let 4a+, be the least ordinal such that

ga+, e HzJ Continuing in this fashion, the construction can be completed so that conditions (i), (ii) and (iii) hold.

Now define a map S: i' -> [She by S (a) = Sa where Sa is defined as follows: For each finite partial isomorphism p: F - F with dom p invariant under ga, by (iii) we can choose /lso > a so that we have /lso V U,<a Sb and APga p1 (a) = ggp (a) for all a e ran p. Let

Sa= {/| dom p is invariant under ga }.

Note that for a & ,fl, Sa n Sp 0. Let F be the partial order consisting of all conditions (p, F) such that:

(1) p : F - F is a finite partial isomorphism (2) F n - i is a finite invective partial map (3) For all a e domF, F(a) e Sa (4) For all a e dom F, dom p is invariant under ga. (5) For all a e dom F and a e ran mp, Ogaf -1 (a) = gF(a) (a)

his ordered by (j1,F1) < (p2,F2) iffp1 D D?2 andF1 D F2.




CLAIM 1. P is a c.c.c. partial order.

PROOF. To prove that P is c.c.c., suppose that { (o,, Fe) < col } is an antichain. Without loss of generality, is, ( o for all 4. Let A, = dom Fe. By the A-system lemma, we may assume that {A, I 4 < co, } forms a A-system with root R. For each a e R, we must have F. (a) e Sa. Thus there are only countably many possibilities for F. [ R. Therefore there is an uncountable X C co, such that F. [ R are all the same for all 4 e X. It follows that {((p, F.:) I 4 e X} are pairwise compatible, a contradiction. -1

CLAIM 2. For each a e F, the set Da {(o, F) e P I a e dom p} is dense.

PROOF. Let ((O, F) e P be arbitrary. Suppose domE { a..a}. Extend (o to h e G. Then (ghl,... ,gh ) is generic and agrees with (gF(a1). gF( ()) on ran po. By Proposition 2.4 (F, gh * .gh) (, gF(F 1l). XgF(an))- Since this structure is homogeneous with respect to sn and the generic sequences agree on ran (p, we can find an automorphism k e Gran s such that for each i 1, n,

ghik = (ai). Let V = hk. Then V extends A, and for each i 1,... n, VI-

gai - gF(i) . Since the sequence (gal .., gan ) is generic we can extend dom p U {a } to a finite set Y such that Y is invariant under gal, gan . Let -c = / Y. Then (OF) < (po,F). -d

CLAIM 3. For each a e F, the set Ea {(O, F) E F a e ranp} is dense.

PROOF. Define fV as before. Extend dom p U { V -1 (a) } to a finite set Y such that Y is invariant under ga,... ., gan. Let - = [v Y. Then (E, F) < (p, F). -1

CLAIM 4. For each a < X, the set Ka = {(, F) e F a a e domF} is dense.

PROOF. Let (, F) e F be arbitrary. Suppose dom F = {oal, . . , an }, and define fV as before. Since the sequence (gal,...gang) is generic we can extend dom p to a finite set Y such that Y is invariant under gal, .. , gag. Let -c = V/ Y. By the definition of S, we can find an appropriate fl > a so that Tga T 1 ran - = gp [ ran -. So (m, F U (a, fl)) < (p, F). -

Now let 9 be a filter intersecting each Da, Ea and Ka. Let

T UP J I (Ro, F) e} and (X J = U{F I(Wo, F) eE }.

By Claim 4, FD K > -K is a strictly increasing function. By Claims 2 and 3, P e Aut(F) = G; and clearly TgaT-1 = gD(a) for each a < S. Since ' e G, we have that ' e Ha, C Hz,, for some a < S. But then we have

go(a) = Tga 1 E eHey,

which is a contradiction. -

REFERENCES

[1] D. EVANS, Examples of co-categorical structures, Automorphisms offirst order structures (R. Kaye and D. Macpherson, editors), Oxford University Press, Oxford, 1994, pp. 33-72.

[2] W HODGES, I. HODKINSON, D. LASCAR, and S. SHELAH, The small index property for co-stable co-categorical structures andfor the random graph, Journal of the London Mathematical Society, vol. 48 (1993), no. 2, pp. 204-218.



1446 STEVE WARNER

[3] E. HRUSHOVSKI, Extending partial automorphisms of graphs, Combinatorica, vol. 12 (1992), pp. 411-416.

[4] A. S. KECHRIS, Classical descriptive set theory, Springer-Verlag, New York, 1995. [5] D. LASCAR, On the category of models of a complete theory, this JOURNAL, vol. 47 (1982), pp. 249-

266. [6] H. D. MACPHERSON and P. M. NEUMANN, Subgroups of infinite symmetric groups, Journal of the

London Mathematical Society, vol. 42 (1990), no. 2, pp. 64-84. [7] J. D. SHARP and S. THOMAS, Uniformization problems and the cofinality of the infinite symmetric

group, Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 328-245. [8] , Unbounded families and the cofinality of the infinite symmetric group, Archive of Mathe-

matical Logic, vol. 34 (1995), pp. 33-45. [9] S. THOMAS, The cofinalities of the infinite dimensional classical groups, Journal of Algebra, vol. 179

(1996), pp. 704-719. [10] J. K. TRUSS, Generic automorphisms of homogeneous structures, Proceedings of the London Math-

ematical Society, vol. 65 (1992), no. 3, pp. 121-141.

DEPARTMENT OF MATHEMATICS RUTGERS UNIVERSITY

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