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.
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
reports proofs of the second Kahn–Kalai conjecture , which bounds graph-containment thresholds using subgraph expectations, and of Talagrand’s discrete convexity conjecture , both discussed in my account of Jinyoung Park’s ICM lecture . A companion result makes integral and fractional expectation…