[Paper Review] Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

Noé de Rancourt|arXiv (Cornell University)|May 13, 2018

Advanced Topology and Set Theory21 references4 citations

TL;DR

This paper develops a general framework for infinite-dimensional Ramsey theory without relying on the pigeonhole principle, unifying Gowers' theorem on block sequences in Banach spaces and Borel determinacy. It proves the adversarial Ramsey principle for Borel sets, a conjecture by Rosendal, establishing a new duality between strategies in infinite games and homogeneous subspaces in topological Ramsey theory.

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.

Motivation & Objective

• To develop a general formalism for infinite-dimensional Ramsey theory that includes both results with and without the pigeonhole principle.
• To unify Gowers’ Ramsey-type theorem for block sequences in Banach spaces and Borel determinacy of games on integers.
• To prove the adversarial Ramsey principle for Borel sets, a conjecture by Rosendal.
• To extend the scope of Ramsey-theoretic methods beyond classical forcing and pigeonhole-based arguments.
• To establish a connection between strategic determinacy in infinite games and the existence of homogeneous subspaces in topological spaces.

Proposed method

• Introduces a general framework for infinite-dimensional Ramsey theory using the concept of 'approximate' and 'exact' games, inspired by Gowers’ and Rosendal’s work.
• Defines Gowers-type games on structured spaces (e.g., block subspaces in Banach spaces) and analyzes player strategies in these games.
• Applies a system of precompact sets to model 'approximate' subspaces, enabling compactness arguments even when full subspaces are not available.
• Uses the notion of $Δ$-expansion of a set to define approximate homogeneous sets, crucial for handling non-pigeonhole settings.
• Applies determinacy techniques to show that if player I has a winning strategy in a game, then there exists a homogeneous subspace in the $Δ$-expansion of the target set.
• Leverages Rosendal’s exact version of Gowers’ theorem and extends it to Borel sets via a game-theoretic duality.

Experimental results

Research questions

• RQ1 Can infinite-dimensional Ramsey theory be developed without relying on the pigeonhole principle, particularly in settings like Banach spaces?
• RQ2 Is there a unifying principle that generalizes both Gowers’ theorem on block sequences and Borel determinacy of games on integers?
• RQ3 Under what conditions does the existence of a winning strategy for player I in a Gowers-type game imply the existence of a homogeneous subspace in a $Δ$-expansion?
• RQ4 Can the adversarial Ramsey principle be proven for Borel sets in general topological Ramsey spaces?
• RQ5 What structural conditions on the space and the target set ensure that a strategy in a game implies homogeneity in a perturbed sense?

Key findings

• The adversarial Ramsey principle for Borel sets is proven, confirming a conjecture by Rosendal.
• The framework successfully handles Ramsey-type results without the pigeonhole principle, extending beyond classical Mathias–Silver-type theorems.
• A new duality is established between winning strategies in infinite games and the existence of homogeneous subspaces in $Δ$-expansions of Borel sets.
• The result generalizes both Rosendal’s exact version of Gowers’ theorem and Borel determinacy of games on integers.
• Counterexamples show that the conclusion fails in infinite fields even when player I has a strategy, highlighting the necessity of compactness and finiteness in the framework.
• The proof relies on a system of precompact sets and compactness arguments, showing that 'approximate' subspaces can still yield homogeneous structures.