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The Ascoli-He-Park-Talagrand graph-decomposition conjecture at the expectation threshold

For a graph HH on at most nn vertices let pc(H;n)p_c(H;n) be the least pp at which G(n,p)G(n,p) contains a copy of HH with probability 1/21/2, and q(H;n)q(H;n) its integral expectation threshold. The Park-Pham theorem gives pc(H;n)≤Cq(H;n)log⁡∣E(H)∣p_c(H;n)\le Cq(H;n)\log|E(H)|, and the logarithm cannot be dropped in general. Ascoli, He, Park and Talagrand (2026, Conjecture 8.1) proposed that it can be removed after cutting HH into boundedly many pieces, and proved bounded-degeneracy cases. Are there absolute constants kk and LL such that the edges of every graph HH on at most nn vertices split into kk subgraphs H1,…,HkH_1,\dots,H_k with pc(Hi;n)≤L q(H;n)p_c(H_i;n)\le L\,q(H;n) for each ii?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Random graphs; thresholds for subgraph containment
Posed by
Ruben Ascoli, Xiaoyu He, Jinyoung Park and Michel Talagrand (Conjecture 8.1, arXiv 2608.11183, August 2026)
Year posed
2026
Years open
0y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there are absolute k≥2k\ge2 and L≥1L\ge1 such that for every n≥2n\ge2 and every graph HH on at most nn vertices, E(H)=E(H1)⊔⋯⊔E(Hk)E(H)=E(H_1)\sqcup\dots\sqcup E(H_k) with pc(Hi;n)≤Lq(H;n)p_c(H_i;n)\le Lq(H;n) for each ii. Pieces may be empty or share vertices, and the partition depends only on HH and nn. It does not give a single copy of HH at threshold Lq(H;n)Lq(H;n) (the separate embeddings need not glue), and kk, LL are not explicit.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof uses the family's discrete-convexity theorem (September 23, 2026) as an essential input; the manuscript says so.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Conjecture 8.1 as the manuscript reports it; the manuscript notes that the conjecture's overlapping-edge-set form is equivalent to the partition form proved. The partition is fixed before sampling, and the embeddings of different pieces need not agree on shared vertices. The proof is conditional on the companion discrete-convexity theorem, itself unreviewed. No Lean formalization of this manuscript.

Sources

Changelog1 change

Discussion