Determinacy of adversarial Gowers games

TL;DR

This paper establishes a game theoretic dichotomy for certain sets of block sequences in vector spaces, extending Gowers' block Ramsey theorem and Davis' determinacy results for $G_{\delta\sigma}$ sets.

Contribution

It introduces a new dichotomy for $G_{\delta\sigma}$ sets of block sequences, bridging combinatorial and determinacy results in vector spaces.

Findings

Proves a dichotomy for $G_{\delta\sigma}$ sets of block sequences.

Extends Gowers' block Ramsey theorem to broader classes.

Connects combinatorial and game-theoretic perspectives.

Abstract

We prove a game theoretic dichotomy for $G_{\delta\sigma}$ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis' proof of $G_{\delta\sigma}$ determinacy.