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 an infinite-dimensional combinatorial subspace, i.e. another copy of [math][2]^\omega[/math].

The generalization of this theorem to higher k is the Carlson-Simpson theorem.