Ramsey's theorem

English

Etymology

Named after British mathematician and philosopher Template:W.

Noun

1. Template:Lb A (version of a) theorem concerning the existence of cliques in a labelled complete graph.
   1. The theorem that any Template:W (with colours) of a sufficiently large complete graph contains monochromatic cliques.
   2. The theorem that any Template:W (with colours) of an infinite complete graph contains at least one infinite monochromatic clique.

Usage notes

Equivalent statements exist for other mathematical contexts. For instance, for a combinatoric statement of the infinitary version: If $A_1\cup A_2\cup \dots \cup A_n={\mathbb{N}}^2$ is a partition of ${\mathbb{N}}^2$, then there exists an infinite subset $X\subseteq \mathbb{N}$ that is homogeneous for the partition (i.e., for some $i,X^2\subseteq A_i$).

Synonyms

• Template:Sense finite Ramsey's theorem
• Template:Sense infinite Ramsey's theorem

Further reading

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