Carlson-Simpson theorem

Carlson-Simpson theorem (k=3): If [math]\displaystyle{ [3]^\omega := \bigcup_{n=0}^\infty [3]^n }[/math] is partitioned into finitely many color classes, then one of the color classes contains an infinite-dimensional combinatorial subspace, i.e. another copy of [math]\displaystyle{ [3]^\omega }[/math].

Implies the coloring Hales-Jewett theorem. The k=2 version already implies Hindman's theorem.

The Carlson-Simpson theorem and the Graham-Rothschild theorem have a common generalisation, Carlson's theorem.

Both the Carlson-Simpson theorem and Carlson's theorem are is used in the Furstenberg-Katznelson argument.