Some recent results in Ramsey theory

TL;DR

This survey surveys sharp advances in graph Ramsey theory, focusing on diagonal, off-diagonal, induced, and multicolour Ramsey numbers. It highlights a sequence of breakthroughs across probabilistic constructions (e.g., triangle-free processes and Kim’s nibble), blow-up and random-graph methods, and the novel use of hypergraph containers to handle induced-structure phenomena. A centerpiece is the exponential improvement for diagonal Ramsey numbers $R(k)$ and the near-diagonal refinements that push the envelope for fixed \ell$ and large $k$, as well as the first substantial lower bounds for $R(4,k)$ and the induced Ramsey framework by ACDFM. The collective methodological theme combines random graphs, pseudorandomness, and container techniques to yield more precise asymptotics and new mechanisms for controlling independent sets and clique structures, with implications for combinatorics, probability, and theoretical computer science.

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.

Figures (2)

• Figure 7.1: The setting of the proof of Theorem \ref{['thm:GNNW']}.
• Figure 9.1: The setting of the Multicolour Book Algorithm.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Erdős and Szekeres, 1935
• proof
• Theorem 2.2: Shearer, 1983
• proof : Sketch of the proof
• proof : Proof of the upper bound in Theorem \ref{['thm:R3k']}