The second Kahn–Kalai conjecture

We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for $G(n,p)$ to contain an ordinary copy of H is at most $C{p}_{\mathrm{E}}(n,H)(1+{\log }_{2}h)$, where C is universal. Here $p_{\mathrm{E}}(n,H)$ is the least density at which every subgraph of H has expected copy count at least one half.