Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

TL;DR

This paper introduces a new framework for infinite-dimensional Ramsey theory that extends existing results by removing the pigeonhole principle and proves the adversarial Ramsey principle for Borel sets, unifying several key concepts.

Contribution

It develops a general framework for infinite-dimensional Ramsey theory without relying on the pigeonhole principle and proves the adversarial Ramsey principle for Borel sets, confirming a conjecture by Rosendal.

Findings

Proved the adversarial Ramsey principle for Borel sets.

Unified Gowers' theorem and Borel determinacy within a new framework.

Extended Ramsey theory to infinite dimensions without pigeonhole assumptions.

Abstract

We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers' Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers' theorem and Borel determinacy of games on integers.