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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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