Some recent results in Ramsey theory

TL;DR

This survey reviews recent advances in graph Ramsey theory, highlighting exponential improvements in Ramsey numbers, new lower bounds, and bounds on induced Ramsey numbers, showcasing significant progress in the field.

Contribution

It summarizes recent breakthroughs and proof techniques that have led to exponential improvements and new bounds in various Ramsey number problems.

Findings

Exponential improvements to diagonal, near-diagonal, and multicolour Ramsey numbers

Improved lower bounds on R(3,k) and R(4,k)

Exponential upper bound on induced Ramsey numbers

Abstract

The purpose of this survey is to provide a gentle introduction to several recent breakthroughs in graph Ramsey theory. In particular, we will outline the proofs (due to various groups of authors) of exponential improvements to the diagonal, near-diagonal, and multicolour Ramsey numbers, improved lower bounds on $R(3,k)$ and $R(4,k)$, and an exponential upper bound on the induced Ramsey numbers.