A new lower bound for the Ramsey numbers $R(3,k)$

We prove a new lower bound for the off-diagonal Ramsey numbers, $$R(3,k)\ge (\frac{1}{3}+o(1))\frac{k^2}{\log k},$$ thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves the best known lower bound of $(1\mathrm{/}4+o(1))k^2\mathrm{/}\log k$ due, independently, to Bohman and Keevash, and Fiz Pontiveros, Griffiths and Morris, resulting from their celebrated analysis of the triangle-free process. As a consequence, we disprove a conjecture of Fiz Pontiveros, Griffiths and Morris that the constant $1\mathrm{/}4$ is sharp.