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

What OpenAI claims

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_E(n,H)(1+log₂ h), with universal C. Here p_E is the least density at which every subgraph of H has expected copy count at least 1/2. (Lean)

Paper

1. The second Kahn–Kalai conjecture

   24 September 2026 · Preprint, not peer reviewed · PDF

   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_E(n,H)(1+log₂ h), where C is universal. Here p_E(n,H) is the least density at which every subgraph of H has expected copy count at least one half.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.