Hales-Jewett type configurations in small sets

TL;DR

This paper demonstrates that Hales-Jewett type configurations exist in small Ramsey-theoretic sets using elementary methods, extending previous algebraic results to more general contexts.

Contribution

It shows the presence of Hales-Jewett configurations in small sets like J-sets and C-sets through elementary proofs, broadening the scope of prior algebraic findings.

Findings

Configurations exist in small Ramsey-theoretic sets

Elementary proof techniques are effective in this context

Extends results from piecewise syndetic sets to smaller sets

Abstract

In a recent work, N. Hindman, D. Strauss and L. Zamboni have shown that the Hales-Jewett theorem can be combined with a sufficiently well behaved homomorphisms. Their work was completely algebraic in nature, where they have used the algebra of Stone-Cech compactification of discrete semigroup. They have proved the existence of those configurations in piecewise syndetic sets, which is a Ramsey theoretic rich set. In our work we will show those forms are still present in very small but Ramsey theoretic sets, (like J-set, C-set) and our proof is purely elementary in nature.