Recent Progress in Ramsey Theory

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 Erd\H{o}s-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 $r(4,t) = \Omega(t^3/(\log t)^4)$ as $t \rightarrow \infty$, answering an old conjecture of Erd\H{o}s, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems.