An abstract formulation of image partition regularity

TL;DR

This paper introduces an abstract framework for image partition regularity, extending classical results to more general systems using a variant of the first entry condition, particularly focusing on the Milliken-Taylor system for infinite cases.

Contribution

It presents a novel abstract formulation of image partition regularity and adapts the Milliken-Taylor system for infinite cases, expanding the theoretical understanding.

Findings

Established an abstract version of image partition regularity.

Extended results to infinite systems using Milliken-Taylor system.

Utilized a variant of the first entry condition for proofs.

Abstract

Inspired by the paper [1] of V. Bergelson, John H.Johnson Jr., J. Moreira, we formulate an abstract version of image partition regularity. To establish the result we have used a variant of first entry condition and for infinite case we contained our work to Milliken-Taylor system.