Ramsey properties of random graphs and Folkman numbers

TL;DR

This paper refines the understanding of the threshold for Ramsey properties in random graphs, providing a more explicit bound and a new upper bound on Folkman numbers, with implications for graph coloring and combinatorics.

Contribution

It offers a self-contained proof with improved bounds for the Ramsey threshold in 2-colorings and derives a double exponential upper bound on Folkman numbers, improving previous results.

Findings

Double exponential dependencies in the Ramsey threshold theorem

A new self-contained proof for the 2-color case

Double exponential upper bound on Folkman numbers

Abstract

For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for the Ramsey property $G(n,p)\to (F)_r$, where $G(n,p)$ is a random graph obtained by including each edge of the complete graph on $n$ vertices, independently, with probability $p$. The original proof was based on the regularity lemma of Szemer\'edi and this led to tower-type dependencies between the involved parameters. Here, for $r=2$, we provide a self-contained proof of a quantitative version of the Ramsey threshold theorem with only double exponential dependencies between the constants. As a corollary we obtain a double exponential upper bound on the 2-color Folkman numbers. By a different proof technique, a similar result was obtained independently by Conlon and Gowers.