Partition Theorems for Spaces of Variable Words

Furstenberg and Katznelson applied methods of topological dynamics to Ramsey theory, obtaining a density version of the Hales-Jewett partition theorem. Inspired by their methods, but using spaces of ultrafilters instead of their metric spaces, we prove a generalization of a theorem of Carlson about variable words. We extend this result to partitions of finite or infinite sequences of variable words, and we apply these extensions to strengthen a partition theorem of Furstenberg and Katznelson about combinatorial subspaces of the set of words.