The asymptotics of $r(4,t)$

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 $$r(4,t)=\mathrm{\Omega }\left(\frac{t^3}{{\log }^4\text{\hspace{0.17em}⁣}t}\right)ast\to \mathrm{\infty }$$ which determines $r(4,t)$ up to a factor of order ${\log }^2\text{\hspace{0.17em}⁣}t$, and solves a conjecture of Erdős.