Combining Extensions of the Hales-Jewett Theorem with Ramsey Theory in Other Structures

Abstract

The Hales-Jewett Theorem states that given any finite nonempty set ${\mathbb A}$ and any finite coloring of the
free semigroup $S$ over the alphabet ${\mathbb A}$ there is a variable word over ${\mathbb A}$ all of whose instances are the same color. This theorem has some extensions involving several distinct variables occurring in the variable word. We show that, when combined with a sufficiently well behaved homomorphism,  the relevant variable word simultaneously satisfies a Ramsey-Theoretic conclusion in the other structure. As an example we show that if $\tau$ is the homomorphism from the set of variable words into the natural numbers which associates to each variable word $w$ the number of occurrences of the variable in $w$, then given any finite coloring of $S$ and any infinite sequence of natural numbers, there is a variable word $w$ whose instances are monochromatic and $\tau(w)$ is a sum of distinct members of the given sequence.

Our methods rely on the algebraic structure of the Stone-Čech compactification of $S$ and the other semigroups that we consider.  We show for example that if $\tau$ is as in the paragraph above, there is a compact subsemigroup $P$ of $\beta{\mathbb N}$ which contains all of the idempotents of $\beta{\mathbb N}$ such that, given any $p\in P$, any $A\in p$, and any finite coloring of $S$, there is a variable word $w$ whose instances are monochromatic and $\tau(w)\in A$.

We end with a new short algebraic proof of an infinitary extension of the Graham-Rothschild Parameter Sets Theorem.