# Carlson's theorem

Carlson's theorem (k=3), Version I: If $[4]^\omega := \bigcup_{n=0}^\infty [4]^n$ 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.
Carlson's theorem (k=3), Version I: If the combinatorial lines in $[3]^\omega := \bigcup_{n=0}^\infty [3]^n$ 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 $[3]^\omega$ as an element in $[4]^\omega$ containing at least one 4, thinking of the 4 as the "wildcard" for the line.