A variant of the Hales-Jewett Theorem

TL;DR

This paper extends the Hales-Jewett theorem to show that in any finite partition of positive integers, there exists a cell containing complex configurations combining arithmetic and geometric progressions, generalizing previous results on structured sets.

Contribution

It introduces a Hales-Jewett type extension that guarantees the presence of intertwined arithmetic and geometric progressions within partition cells.

Findings

Any partition cell contains configurations with arithmetic and geometric progressions.

The result generalizes previous work on structured sets in positive integers.

Provides a new combinatorial framework for understanding multiplicative and additive structures.

Abstract

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j \in\nhat k}\subset B. $ In particular one cell of each finite partition of the positive integers contains such configurations. We prove a Hales-Jewett type extension of this partition theorem.