Submitted by:

Suparno Ghoshal (Roll No.: 133)

Neelabja Roy (Roll No.: 135)

Indayan Bera (Roll No.: 134)

Tanwi Roy (Roll No.: 132)

Dept: C.S.E

Sec.: A

4/2

5/2

015

1

Grap

h th

eory

pro

ject 2nd

SE

M C

SE

201

5

4/2

5/2

01

5

2

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

Introduction

Ramsey Theorem

Problems Leading to Ramsey Theorem

Generalized Ramsey Theory Involving Formation of Monochromatic Triangles &

It’s Proof

Several Relations Involving Ramsey & Proofs of the Above Relations

Shur Problem

Schur Problem and Ramsey Theory

Solution of the Schur Problem Involving Ramsey Theory

Conclusion

References

OUTLINE

4/2

5/2

01

5

3

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

Introduction Ramsey's theorem is a foundational result in combinatorics.

Ramsey theory, seeks regularity amid disorder: general conditions for the existence of

substructures with regular properties.

In this application it is a question of the existence of monochromatic subsets, that is,

subsets of connected edges of just one color.

The generalized Ramsey Theorem is still an unsolved problem.

So we will be discussing a few special cases of Ramsey Theorem : Ramsey Theory

involving the formation of monochromatic triangles.

We would also show the lower and upper bounds of a Ramsey Number.

We would use Paul Erdös’s inequality to find the closed form of the upper bound of

Ramsey Number.

We would also touch upon Schur’s problem which is a Number Theoretic extension of the

Ramsey Theory dealing with the partition of sets into sum-free subsets.

The Schur’s problem would be used to derive the lower bound of Ramsey Number.

4/2

5/2

01

5

4

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

Problems LEADING to Ramsey Theorem :

Among six persons, there are always three who know

each other or three who are complete strangers.

(This problem was proposed in 1947 in the Kürschak

Competition and in 1953 in the famous Putnam

Competition)

Graph for problem 1

Each of 17 scientists corresponds with all others. They

correspond about only three topics and any two treat exactly

one topic. Prove that there are at least three scientists, who

correspond with each other about the same subject.

4/2

5/2

01

5

5

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

If q1, q2,......,qn ≥ 2 are integers, then there is

a minimal number R(q1, q2,......,qn), so that, for

p ≥ R(q1, q2,......,qn ) for at least one i =1,....,n,

Gp contains at least one monochromatic Gqi.

where, Gp: A complete graph consisting of p

vertices.

Ramsey Theorem :

4/2

5/2

01

5

6

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

In space, there are given pn = [en!] + 1 points. Each pair of points is

connected by a line, and each line is colored with one of n colors. Prove

that there is at least one triangle with sides of the same color.

The first two problems are special cases of the third with n=2 and

n=3.one represents the persons by points.

In the first problem, each pair of points is joined by red or blue segment

depending on the corresponding persons being acquaintances or strangers.

In the second problem each pair of points is joined by a red, blue or green

segment if the corresponding scientists exchange letters about the first,

second, or third topic respectively.

Generalized Ramsey theory (considering only monochromatic triangles):

PROOF: 4

/25

/20

15

7

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

In general we get :

( pn+1-1) / (n+1) = ( pn-1)/(1+(1/(n+1) ));

Let qn=pn-1, we get,

qn+1=(n+1)qn+1

Dividing by (n+1)!,

qn+1/(n+1)!=qn/n!+1/(n+1)!

From this we can easily get

qn = n!(1+1/1!+1/2!+....+1/n!)

Recognize the fact that the second part in parenthesis is the e series.

e= qn n! + rn;

rn=1/(n+1)!+1/(n+2)!+....<1/n!(1/(n+1)+1/(n+1)!+..)=1/ (n*n!)

Hence,

qn <en! < qn +1/n,

i.e,

qn =[en!] or pn=[en!]+1.

4/2

5/2

01

5

8

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

Several relations involving Ramsey theory:

Upper Bound of R(r,s): There are a couple of estimates given by

Erdös giving the upper bound of R(r,s)

i) R(r,s) ≤ R(r-1,s) + R(r,s-1)

ii) R(r,s) ≤ r+s-2 Cr-1

Lower Bound of Rn(3): Till now no closed form of the lower bound of

R(r,s) has been given. So we will be discussing the lower estimate of

Rn(3).We will derive the lower estimate of Rn(3) using Schur’s problem.

4/2

5/2

01

5

9

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

PROOF : [Upper Bound of R(r,s)]

i) We consider the complete graph with R(r-1,s) +

R(r,s-1) vertices whose edges are coloured red and black. We select one vertex v and

consider

V1= set of all vertices, which are connected to v by a red edge. │V1│= n1.

V2= set of all vertices, which are connected to v by a black edge.│V2│= n2.

n1 + n2 +1 = R(r-1,s) + R(r,s-1).

From n1< R(r,s-1), we conclude that n2≥ R(r,s-1). This implies that V2 contains a Gr or Gs-

1, and together with v, we have a Gs.

Again n1≥ R(r-1,s) implies that V1 contains a Gs or a Gr-1, and together with v , a Gr.

Thus, we have,

R(r,s) ≤ R(r-1,s) + R(r,s-1).

with the boundary conditions R(2,s) = s, R(r,2) = r. For symmetry reasons, we have

R(r,s) = R(s,r).

ii) We will prove this second inequality using Method of Mathematical Induction.

Considering the base case at r=s=2,

R(2,2) = 2 ≤ 2+2-2C2-1 = 2

Now we proceed using double induction on r and s.

Assume the expression holds for R(r-1,s) and R(r,s-1).

Then

R(r,s) ≤ R(r-1,s) + R(r,s-1) ≤ r+s-3 C r-2 + r+s-3 Cr-1 = r+s-2 C r-1

Hence proved.

4/2

5/2

01

5

10

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

The schur problem :

In connection with the Fermat conjecture, in 1916 Isai Schur

considered the following problem:

►What is the largest positive integer f(n) so that the set

{1,2,...,f(n)}can be split into n sum-free subsets?

schur problem and Ramsey theory :

►Now with connection to the Ramsey theory we modify Schur’s

problem by setting f(n)=[en!].

►We try to show that each partition of the set {1, 2,..., [en!]} into n

subsets has at least one subset in which the equation x+y = z is solvable,

in which x, y, z {1,2,...,[en!]}.

4/2

5/2

01

5

11

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

SOLUTION :

Suppose

{1, 2,..., [en!]}= A1 ՍA2 Ս... ՍAn

is a partition into n parts. We consider the complete graph G with [en!]+1

points, which we label 1, 2,..., [en!]+1. We colour G with n colours 1, 2,..,

n. The edge rs gets colour m, if │r-s│ Am. According to problem1, G

will have a monochromatic triangle, that is there exist positive integers r, s,

t such that r<s<t≤ [en!]+1, so that the edges rs, rt, st all have the same

colour m, that is,

s-r, t-s, t-r Am

.

Because (s-r) + (t-s) = t-r, Am is not sum-free. q.e.d.

4/2

5/2

01

5

12

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

Recalling Rn(3) :

This is the smallest positive integer such that every n-coloring of the complete graph with Rn(3)

vertices forces a monochromatic triangle.

Now before estimating the lower bound of Rn(3) we would try and estimate a lower bound of the

Schur function.

Let f(n) be a Schur function.

We now try and show that Rn(3) ≥ f(n) + 2.

Let A1, A2,...., An be sum-free partition of {1,2,...,f(n)}and suppose that G is a complete graph

with f(n) + 1 vertices 0,1,2,...,f(n).

We colour the edges of G with n colours 1,2,..,n by colouring edge rs with colour m if │r-s│ Am.

Suppose we get a triangle with vertices r, s, t and with edges of colour m. We assume r<s<t . Then

we have t-s, t-r, s-r Am. But (t-s) + (s-r) = t-r, and this contradicts that Am is sum-free. Hence Rn >

f(n) + 1, q.e.d.

Estimate of lower bound:

4/2

5/2

01

5

13

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

Now we try and show that f(n) ≥ (3n – 1)/2.

If the table with n rows

x1,x2,..., ...., u1,u2,...has sum-free rows, then the n+1 rows

3x1, 3x1-1, 3x2, 3x2-1,.....1,4,7,...., 3f(n) + 1

In any case f(n+1) ≥ 3f(n) + 1, and since we have f(1)=1, f(2) ≥ 4, f(3) ≥ 13.

Thus we get f(n) ≥ 1+3+32+33+...+3n-1=(3n -1)/2.

Therefore: Rn(3) ≥ (3n + 3)/2

Estimate of lower bound: (contd..)

4/2

5/2

01

5

14

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

♠ Several research work has been carried out in the field of Graph Theory

involving Ramsey Theorem.

♠ But the generalized Ramsey Theorem is still an open problem.

♠ In this project we tried to merge ideas of graph theory and number theory.

♠ We showed the connection between Ramsey Theory and Schur’s theory.

♠ Schur’s Theory, which is a topic of number theory, involving the partition of

sets into sum-free subsets, has a very elegant relation with the Ramsey

Theorem which is a topic of graph theory.

Conclusion

4/2

5/2

01

5

15

Grap

h th

eory

pro

ject 2nd S

EM

CS

E 2

015

References:

♣ Applied Combinatorics by Alan Tucker.

♣ Problem Solving Strategies by Arthur Engel

♣ Graph Theory by N.Deo

♣ wolfram mathworld.com