Phase transition results for three Ramsey-like theorems

TL;DR

This paper establishes a precise phase transition threshold for Friedman's finite adjacent Ramsey theorem and extends the methodology to related Ramsey variants, providing tools to streamline such proofs.

Contribution

It introduces a unified approach to classify phase transitions in Ramsey-like theorems and enhances proof techniques by reducing ad-hoc arguments.

Findings

Classified the phase transition threshold for Friedman's theorem

Extended the classification method to Paris--Harrington and Kanamori--McAloon theorems

Provided tools to simplify proofs of phase transition results

Abstract

We classify a sharp phase transition threshold for Friedman's finite adjacent Ramsey theorem. We extend the method for showing this result to two previously known classifications involving Ramsey theorem variants: the Paris--Harrington theorem and the Kanamori--McAloon theorem. We also provide tools to remove ad-hoc arguments from the proofs of phase transition results as much as currently possible.