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 all elements of an infinite-dimensional combinatorial subspace which contain the digit 1, and such that none of the fixed digits of this subspace are equal to 1.

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