Hindman's theorem

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Hindman's theorem: If [math][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 which replaces 2 with larger k is Carlson's theorem. Hindman's theorem also implies Folkman's theorem.