The asymptotics of $r(4,t)$

Abstract

For integers $s,t\ge 2$, the Ramsey numbers $r(s,t)$ denote the minimum $N$ such that every $N$-vertex graph contains either a clique of order $s$ or an independent set of order $t$. In this paper we prove \[ r(4,t) = Ω\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty\] which determines $r(4,t)$ up to a factor of order ${\log }^{2}t$, and solves a conjecture of Erdős.

Updated journal version