Recent Progress in Ramsey Theory

TL;DR

The paper introduces a spectral/pseudorandom-graph framework for Ramsey theory, connecting eigenvalue structure to Ramsey bounds via transference from extremal to Ramsey problems. Using tools such as the Alon–Boppana bound, the Expander Mixing Lemma, and container results, it derives lower bounds on Ramsey numbers by constructing $F$-free, pseudorandom graphs and transferring extremal properties from hypergraphs. A key achievement is proving $r(4,t) = \Omega\left(\frac{t^3}{(\log t)^4}\right)$, matching Erdős’ conjecture up to polylogarithmic factors, and obtaining near-tight bounds for several cycle- Ramsey numbers (e.g., $r(C_5,t)$). The approach yields wide-ranging implications for hypergraph Ramsey numbers, Erdős–Rogers functions, and Ramsey-minimal graphs, offering a robust toolkit for tackling off-diagonal, diagonal, and hypergraph Ramsey problems with potential broad impact in combinatorics.

Abstract

The classical Ramsey numbers $r(s,t)$ denote the minimum $n$ such that every red-blue coloring of the edges of the complete graph $K_n$ contains either a red clique of order $s$ or a blue clique of order $t$. These quantities are the centerpiece of graph Ramsey Theory, and have been studied for almost a century. The Erdős-Szekeres Theorem (1935) shows that for each $s \geq 2$, $r(s,t) = O(t^{s - 1})$ as $t \rightarrow \infty$. We introduce a new approach using pseudorandom graphs which shows $r(4,t) = Ω(t^3/(\log t)^4)$ as $t \rightarrow \infty$, answering an old conjecture of Erdős, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Conjecture 1
• Conjecture 2
• Theorem 3
• Theorem 4
• Conjecture 3
• Theorem 5
• Theorem 6
• Theorem 7