Ramsey Theory

In General > s.a. Baire Category Theorem; Coloring Problem.
* Idea: A branch of combinatorics or discrete mathematics, consisting of results that are centered around Ramsey's theorem, and, abstractly, attempt to decide when a bipartite graph has the Ramsey property.
* Results: 1990, The only known Ramsey numbers are

R(3,3) = 6,   R(3,4) = 9,   R(3,5) = 14,   R(3,6) = 18,   R(3,7) = 23,   R(3,8) = 29,   R(3,9) = 36,   R(4,4) = 18 .

* Ramsey number for two graphs: For two graphs G₁ and G₂, the Ramsey number R(G₁, G₂) is the smallest integer p such that for any graph G on p vertices either G contains G₁ or G* contains G₂, where G* denotes the complement of G.
@ General references: Graham 80; Graham & Spencer SA(90)jul.
@ Related topics: Huang et al EJC(06) [for two groups, bounds].
> Related topics: see Polytopes.

Ramsey Property for r Colors
$ Def: A bipartite graph has it if for every r-coloring of the class A, there is a monochromatic vertex in B.

Ramsey's Theorem > s.a. partition.
* Idea: A special case of partition relation, for infinite cardinality; Can be generalized to finite sets.
$ Def: The combinatorial set theoretical theorem which states that ₀ → (₀)^k_n, for all finite n and k.
@ References: Ramsey PLMS(30); Erdös & Rado BAMS(56) [generalization].

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