Sharp thresholds for Ramsey properties

TL;DR

This paper introduces a unified framework to establish sharp threshold results for various Ramsey properties by viewing them as non-colourability of hypergraphs, and verifies these conditions in classical and arithmetic settings.

Contribution

It provides a general method to prove sharp thresholds for Ramsey properties across different mathematical contexts, including graphs and number theory.

Findings

Sharp thresholds for graph Ramsey properties in $G_{n,p}$

Sharp thresholds for arithmetic Ramsey properties like van der Waerden and Schur's theorems

Verification of conditions in multiple cases of interest

Abstract

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as non-colourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this non-colourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices. Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph $H$ in $r$ colours has a sharp threshold in $G_{n,p}$, for all $r \ge 2$ and all $H$ in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden's theorem and Schur's theorem, also in any number of colours.