An analogue to infinitery Hales-Jewett theorem

TL;DR

This paper extends the Hales-Jewett theorem by replacing the alphabet with an increasing sequence, demonstrating new configurations in Ramsey theory and broadening the theorem's applicability.

Contribution

It introduces a novel extension of the Hales-Jewett theorem using increasing sequences of alphabets and small Ramsey-theoretic sets, with new configurations.

Findings

Extension of Hales-Jewett theorem to increasing alphabet sequences

Identification of configurations in Ramsey-theoretic small sets

Generalization of combinatorial structures

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. In this paper we will show that those combined extensions can be made if we replace the alphabet by an increasing sequence of alphabets, infact it holds for some Ramsey theoretic small sets. To obtained this we achieved some interesting configurations.