Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

TL;DR

This paper proves sharp thresholds for the Ramsey property in random graphs for a broad class of graphs including cycles, using simplified methods based on Friedgut's criteria and the container method.

Contribution

It provides a simpler proof of sharp thresholds for Ramsey properties in random graphs for nearly bipartite graphs, extending previous results to include all cycles.

Findings

Established sharp thresholds for Ramsey properties in random graphs.

Extended results to include all cycles and nearly bipartite graphs.

Utilized modern combinatorial tools like the container method.

Abstract

For a given graph $F$ we consider the family of (finite) graphs $G$ with the Ramsey property for $F$, that is the set of such graphs $G$ with the property that every two-colouring of the edges of $G$ yields a monochromatic copy of $F$. For $F$ being a triangle Friedgut, R\"odl, Ruci\'nski, and Tetali (2004) established the sharp threshold for the Ramsey property in random graphs. We obtained a simpler proof of this result which extends to a more general class of graphs $F$ including all cycles. The proof is based on Friedgut's criteria (1999) for sharp thresholds, and on the recently developed container method for independent sets in hypergraphs by Saxton and Thomason, and Balogh, Morris and Samotij. The proof builds on some recent work of Friedgut et al. who established a similar result for van der Waerden's theorem.