Carlson-Simpson theorem
From Polymath1Wiki
Carlson-Simpson theorem (k=3): If ![[3]^\omega := \bigcup_{n=0}^\infty [3]^n](/polymath1/images/math/8/4/7/84769445ae7b2a982de43ee196507509.png)
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]ω.
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.
