The asymptotics of r(4,t)

For integers s,t ≥ 2, the Ramsey number $r(s,t)$ denotes the minimum n such that every n-vertex graph contains a clique of order s or an independent set of order t. In this paper we prove \[ r(4,t) = Ω(t^3/log^4 t)as\ t → ∞, \] which determinesr(4,t)$ up to a factor of order $log^2 t$, and solves a conjecture of Erdős.