Hindman's theorem

Hindman's theorem: If [math]\displaystyle{ [2]^\omega := \bigcup_{n=0}^\infty [2]^n }[/math] is finitely colored, then one of the color classes contain an infinite-dimensional combinatorial subspace, i.e. another copy of [math]\displaystyle{ [2]^\omega }[/math].

The generalization of this theorem to higher k is the Carlson-Simpson theorem. Hindman's theorem also implies Folkman's theorem.