An exact Ramsey principle for block sequences

TL;DR

This paper establishes an exact Ramsey principle for infinite block sequences in vector spaces, linking winning strategies in two types of infinite games and recovering Gowers' dichotomy theorem through determinacy.

Contribution

It introduces an exact, non-approximate Ramsey principle for block sequences, connecting game-theoretic strategies to classical dichotomy results in vector spaces.

Findings

Proves an exact Ramsey principle for block sequences

Links strategies in Gowers' game and asymptotic game

Recovers Gowers' dichotomy theorem using determinacy

Abstract

We prove an exact, i.e., formulated without $\Delta$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.