  For integers $s,t \geq 2$, 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 \[ r(4,t) =
\Omega\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow
\infty\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and
solves a conjecture of Erd\H{o}s.
