Ramsey's theorem

Etymology
Named after British mathematician and philosopher.

Noun

 * 1)  A (version of a) theorem concerning the existence of cliques in a labelled complete graph.
 * 2) The theorem that any  (with colours) of a sufficiently large complete graph contains monochromatic cliques.
 * 3) The theorem that any  (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

 * finite Ramsey's theorem
 * infinite Ramsey's theorem