It's a project on graph theory with a few number theoretic extensions of the Ramsey Problem.

1. 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
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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

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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.

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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.

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 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 :

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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):

7. PROOF:
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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.

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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.

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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.

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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!]}.

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SOLUTION :
Suppose
{1, 2,..., [en!]}= A1 Ս A2 Ս... Ս An
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.

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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:

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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..)

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♠ 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

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References:
♣ Applied Combinatorics by Alan Tucker.
♣ Problem Solving Strategies by Arthur Engel
♣ Graph Theory by N.Deo
♣ wolfram mathworld.com