The second Kahn–Kalai conjecture with an edge-count bound

Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in $G(n,p)$ is at most $C p_{\mathrm E}(n,H)(1+\log_2 h)$, with universal C. Here $p_{\mathrm E}$ is the least density at which every subgraph of H has expected copy count at least 1/2.