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 Erdős-Szekeres Theorem (1935) shows that for each $s\ge 2$, $r(s,t)=O(t^{s-1})$ as $t\to \infty$. We introduce a new approach using pseudorandom graphs which shows $r(4,t)=\Omega (t^3/(\log t{)}^4)$ as $t\to \infty$, answering an old conjecture of Erdős, and we illustrate how to apply this approach to many other Ramsey and related combinatorial problems.