Ramsey Theory

Chandler Burfield

April 11, 2013

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Outline

1 IntroductionRamsey TheoryMotivating ExampleDefinitions

2 Ramsey’s Theorem and Ramsey Numbers

3 Applications

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Ramsey Theory

Ramsey theory is a theory that expresses the guaranteed occurrence ofspecific structures in part of a large structure that has been partitionedinto finitely many parts.

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Motivating Example

The Party Problem

In a party of 6 people there will always be a group of 3 people who eitherknow each other or are strangers to each other.

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Defintions

Complete Graph Kn

Kn is a complete graph with n vertices if each pair of vertices in the graphis connected by an edge.

k-coloring

For a graph G , a k-coloring of the edges is any assignment of one of kcolors to each edge of G .

Subgraph

A graph whose vertices and edges are contained within a larger graph.

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Ramsey’s Theorem

Ramsey’s Theorem

Given any positive integers p and q there exists a smallest integern = R(p, q) such that every 2-coloring of the edges of Kn contains either acomplete subgraph of p vertices, all of whose edges are in color 1, or acomplete subgraph of q vertices, all of whose edges are in color 2.

Generalizations

k-colors

Hypergraphs

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Ramsey Numbers

Ramsey Numbers

The integers R(n1, n2, ..., nk) are Ramsey numbers.

Ramsey numbers indicate how big a set must be to guarantee theexistence of certain structures. Relatively few nontrivial Ramsey numbershave been discovered.

”[...] imagine an alien force, vastly more powerful than us, landing on Earth and

demanding the value of R(5,5) or they will destroy our planet. In that case, [...],

we should marshal all our computers and our mathematicians and attempt to find

the value. Suppose, instead, that they ask for R(6,6). In that case, [...], we

should attempt to destroy the aliens.” -Paul Erdos

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Bounds on Diagonal Ramsey Numbers

!

v Kk1�1

Kk1 KR ⇤

R(3, 3) = 6R(3, 3) 6

R(3, 3) = 6 R(3, 3, 3)

R(r, s) r = s

n R(n, n)R(1, 1) = 1R(2, 2) = 2R(3, 3) = 6R(4, 4) = 18

43 R(5, 5) 49102 R(6, 6) 165205 R(7, 7) 540282 R(8, 8) 1870565 R(9, 9) 6588

798 R(10, 10) 23556

R(r, s) R(r � 1, s) + R(r, s � 1).

n = R(r� 1, s)+R(r, s� 1)� 1 Kn Kr

Ks v n � 1R(r � 1, s) � 1 R(r, s � 1) � 1v R(r � 1, s) � 1

v V |V | = R(r � 1, s) � 1V Kr�1 Ks

v VKr

n(R(r � 1, s) � 1)/2

R(r � 1, s) R(r, s� 1)R(r�1, s) R(r, s�1)Kr Ks

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Sketch of Proof of Ramsey’s Theorem

Proof by Induction on p + q.

Base Case: Let p + q = 2. It is clear that R(1, 1) = 1.

Inductive Step: Assume the theorem holds for p + q < N. Prove for P + Q = N.

We know that P + Q − 1 < N and that R(P − 1,Q) and R(P,Q − 1) exist. Consider a2-coloring of the edges of Kv with colors c1 and c2 where the number of verticesv ≥ R(P − 1,Q) + R(P,Q − 1).

Consider vertex x of Kv . By the pigeonhole principle we know that x is incident to eitherR(P-1,Q) edges of color c1 or R(P,Q-1) edges of color c2.

If x is incident to R(P − 1,Q) edges of color c1, consider KR(P−1,Q) whose vertices are joined tox by color c1. We know R(P − 1,Q) exists and must consider two cases. First where KR(P−1,Q)

contains a KP−1 of color c1. In this case, if we add in x we have a monochromatic Kp of colorc1. The second case is that KR(P−1,Q) contains a KQ of color c2. Thus R(P,Q) exists in bothcases.

An analogous argument proves that Kv has one of the required monochromatic complete graphs

in the case that x is incident to R(P,Q-1) edges of color c2. Therefore R(P,Q) exists and we

know R(P,Q) ≤ R(P − 1,Q) + R(P,Q − 1).

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Applications

Convex polygons among points in a plane

Shur’s Theorem

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Convex Polygons Among Points in a Plane

Geometric statement that follows from Ramsey’s theorem for4-uniform hypergraphs

Theorem: For any m ≥ 4, there is n, such that given anyconfiguration of n points in the plane, no three on the same line,there are m points forming a convex polygon.

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Schur’s Theorem

For every r ∈ N there exists a natural number n such that any r -coloringof the natural numbers 1 to n has a monochromatic x , y , and z such thatx+y=z.

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References

Bela Bollobas, Graduate Texts in Mathematics: Modern Graph Theory.Springer-Verlag, 1998.

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