Recent developments in graph Ramsey theory

TL;DR

This survey reviews recent progress in graph Ramsey theory, focusing on understanding the growth and bounds of Ramsey numbers for various graphs, driven by advances in extremal combinatorics.

Contribution

It summarizes recent developments and new results in the quantitative understanding of graph Ramsey numbers and their variants.

Findings

Progress in bounding Ramsey numbers for specific graph classes

New techniques inspired by extremal combinatorics

Open problems and conjectures in the field

Abstract

Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic copy of $H$. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.