# Carlson-Simpson theorem

From Polymath Wiki

**Carlson-Simpson theorem** (k=3): If [math][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][3]^\omega[/math].

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.