# Carlson-Simpson theorem

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

Implies the coloring Hales-Jewett 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.