Gowers' Ramsey theorem for generalized tetris operations

TL;DR

This paper generalizes Gowers' theorem to include all maps from in_k to in_j derived from nondecreasing surjections, extending the theorem's applicability and answering open questions.

Contribution

It introduces a broad generalization of Gowers' theorem for in_k involving all related maps, and unifies it with the Galvin--Glazer--Hindman theorem within layered partial semigroups.

Findings

Proves a generalized Gowers' theorem for in_k with all associated maps.

Answers a question posed by Bartove1 and Kwiatkowska.

Provides a unified framework connecting Gowers' theorem and the Galvin--Glazer--Hindman theorem.

Abstract

We prove a generalization of Gowers' theorem for $\mathrm{FIN}_{k}$ where, instead of the single tetris operation $T:\mathrm{FIN}_{k}\rightarrow \mathrm{FIN}_{k-1}$, one considers all maps from $\mathrm{FIN}_{k}$ to $\mathrm{FIN}_{j}$ for $0\leq j\leq k$ arising from nondecreasing surjections $f:\left\{ 0,1,\ldots ,k+1\right\} \rightarrow \left\{ 0,1,\ldots ,j+1\right\} $. This answers a question of Barto\v{s}ov\'{a} and Kwiatkowska. We also prove a common generalization of such a result and the Galvin--Glazer--Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.