Combining extensions of the Hales-Jewett\\ Theorem with Ramsey Theory\\ in other structures

TL;DR

This paper extends the Hales-Jewett Theorem by combining it with Ramsey Theory in other structures using algebraic methods, demonstrating new monochromatic variable words with additional properties.

Contribution

It introduces a novel approach to combine Hales-Jewett extensions with Ramsey Theory via algebraic structures, providing new results and a short proof of an infinitary Graham-Rothschild theorem.

Findings

Existence of monochromatic variable words with prescribed algebraic properties

Construction of a compact subsemigroup containing idempotents related to the problem

A new algebraic proof of an infinitary Graham-Rothschild Parameter Sets Theorem

Abstract

The Hales-Jewett Theorem states that given any finite nonempty set $\A$ and any finite coloring of the free semigroup $S$ over the alphabet $\A$ there is a {\it variable word\/} over $\A$ 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-\v Cech 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 $\beta\ben$ which contains all of the idempotents of $\beta\ben$ 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.