Finding Product and Sum Patterns in non-commutative settings

TL;DR

This paper extends a partition regularity result to semirings, proving the existence of monochromatic product and sum patterns in non-commutative settings using combinatorial methods.

Contribution

It generalizes Hindman's conjecture to semirings with non-commutative operations, providing a combinatorial proof and exploring non-commutative cases.

Findings

Proved a monochromatic pattern existence in semirings where each element's multiple is piecewise syndetic.

Extended the result from 2-partitions to arbitrary finite partitions.

Provided a combinatorial proof for a version of Szemerédi's theorem in this context.

Abstract

Hindman conjectured that any finite partition of $\mathbb{N}$ has a monochromatic $\{x,y,x+y,xy\}$. Recently, Bowen proved the result for all 2-partition. In this paper, we extend Bowen's result to any semiring $(S,+,\cdot)$ such that $Ss$ is piecewise syndetic for all $s\in S$. As a method, we gave a combinatorial proof for a piecewise syndetic version of Bergerson and Glasscock's IP$_r^*$ Szemer\'edi Theorem, and discussed the case when the operation is not commutative.