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The second Kahn-Kalai conjecture

Claims Theorem 1.1: for every $n\ge2$ and every finite simple graph $H$ with $h\ge1$ edges and at most $n$ vertices, $p_c(n,H)\le\min\{1,2048e^{50}p_E(n,H)(1+\log_2 h)\}$ , hence $p_c(n,H)\le6144e^{50}p_E(n,H)\log_2 n$ . The logarithm depends only on the number of edges of $H$ , which is stronger than the conjectured $\log n$ . The proof builds on Tran's spread-link cores and the Mossel-Niles-Weed-Sun-Zadik resampling, with a tree reduction replacing Tran's proposed coupling conjecture. Constants are explicit but huge; it says nothing about optimal constants or about hypergraph targets.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The second Kahn-Kalai conjecture · Second Kahn-Kalai conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The second Kahn-Kalai conjecture

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No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against the Kahn-Kalai formulation as cited; the paper notes that normalising expected counts at one half instead of one changes $p_E$ by at most a factor two, so the conjecture is unchanged. The proof was not refereed. Lean-checked: the release's formalization catalogue lists ComparatorChallenges/SecondKahnKalai.json with declaration OAI.LeanBlast.SecondKahnKalai.secondKahnKalaiBounds. The statement was read here: for n >= 2 and any finite simple graph H with at least one edge and at most n vertices, the actual containment threshold (infimum over p of probability >= 1/2) is at most min(1, 2048 e^50 p_E (1 + log2 e(H))) and at most 6144 e^50 p_E log2 n, with p_E defined over all subgraphs of H. That is the headline claim. Not rebuilt here.