A Constructive Interpretation of Ramsey's Theorem via the Product of Selection Functions

TL;DR

This paper presents a computational interpretation of Ramsey's theorem using G"odel's Dialectica interpretation and the product of selection functions, offering an alternative to Spector's bar recursion in proof analysis.

Contribution

It introduces a novel proof-theoretic approach to Ramsey's theorem employing the product of selection functions, expanding the toolkit for analyzing mathematical proofs.

Findings

Produced a computational version of Erd ext{"o}s and Rado's proof

Demonstrated the product of selection functions as an alternative to bar recursion

Applied proof theoretic techniques to a classical combinatorial theorem

Abstract

We use G\"{o}del's Dialectica interpretation to produce a computational version of the well known proof of Ramsey's theorem by Erd\H{o}s and Rado. Our proof makes use of the product of selection functions, which forms an intuitive alternative to Spector's bar recursion when interpreting proofs in analysis. This case study is another instance of the application of proof theoretic techniques in mathematics.