Carlson's theorem

Carlson's theorem (k=3), Version I: If [math]\displaystyle{ [4]^\omega := \bigcup_{n=0}^\infty [4]^n }[/math] is partitioned into finitely many color classes, then there exists an infinite-dimensional combinatorial subspace with no fixed coordinate equal to 4, such that every element of this combinatorial subspace with at least one 4 has the same color.

This theorem is a common generalization of the Carlson-Simpson theorem and the Graham-Rothschild theorem. It plays a key role in the Furstenberg-Katznelson argument. It is necessary to restrict to elements containing at least one 4; consider the coloring that colors a string black if it contains at least one 4, and white otherwise.

It has an equivalent formulation:

Carlson's theorem (k=3), Version I: If the combinatorial lines in [math]\displaystyle{ [3]^\omega := \bigcup_{n=0}^\infty [3]^n }[/math] are partitioned into finitely many color classes, then there exists an infinite-dimensional combinatorial subspace such that all combinatorial lines in this subspace have the same color.

This follows by viewing a combinatorial line in [math]\displaystyle{ [3]^\omega }[/math] as an element in [math]\displaystyle{ [4]^\omega }[/math] containing at least one 4, thinking of the 4 as the "wildcard" for the line.

The k=2 version of this theorem is Hindman's theorem.