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Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant

For an increasing family F\mathcal F of subsets of a finite set XX, call F\mathcal F pp-small if some family G\mathcal G with every member of F\mathcal F containing a member of G\mathcal G has ∑S∈Gp∣S∣≤1/2\sum_{S\in\mathcal G}p^{|S|}\le1/2; the expectation threshold q(F)q(\mathcal F) is the largest such pp. The fractional expectation threshold qf(F)q_f(\mathcal F) allows instead weights g(S)∈[0,1]g(S)\in[0,1] with ∑S⊆Hg(S)≥1\sum_{S\subseteq H}g(S)\ge1 for H∈FH\in\mathcal F and ∑Sg(S)p∣S∣≤1/2\sum_S g(S)p^{|S|}\le1/2, so q≤qfq\le q_f. Frankston-Kahn-Narayanan-Park and Park-Pham proved the Kahn-Kalai comparisons with the threshold pcp_c; special cases of Talagrand's comparison were known for weights on singletons, pairs, cliques and bounded-size sets. Talagrand (2010, Conjecture 6.3) asked: is there a universal constant LL with qf(F)≤L q(F)q_f(\mathcal F)\le L\,q(\mathcal F) for every increasing family?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Probabilistic combinatorics; thresholds
Posed by
Michel Talagrand (Conjecture 6.3, 'Are many small sets explicitly small?', STOC 2010)
Year posed
2010
Years open
16y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
38 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: qf(F)≤25⋅5124 q(F)q_f(\mathcal F)\le25\cdot512^4\,q(\mathcal F) for every finite nonempty XX and nonempty proper increasing F⊆2X\mathcal F\subseteq2^X, independently of ∣X∣|X|, the sizes of minimal members and the support of the fractional cover; equivalently every fractional cover of cost at most 1/21/2 at rr rounds to an integral cover of cost at most 1/21/2 at r/(25⋅5124)r/(25\cdot512^4). Previously the best general loss was of order log⁡log⁡∣X∣\log\log|X| (Fischer-Person with Pham). The constant is explicit but not optimized.

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. It belongs to a family with 'Talagrand's discrete-convexity conjecture' (same date) and 'Graph Decompositions at the Integral Expectation Threshold' (October 5, 2026), each answering a different posed problem.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Talagrand's Conjecture 6.3 as the manuscript states it. It gives qf(F)≤25⋅5124 q(F)q_f(\mathcal F)\le25\cdot512^4\,q(\mathcal F) for every nonempty proper increasing family on a finite nonempty ground set, both with budget 1/21/2. lean/formalization.yaml lists TalagrandExpectationThreshold.json (declaration OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence, file OAI/Combinatorics/ExpectationThreshold/Main.lean). The statement TalagrandExpectationThreshold.lean was read here; not rebuilt here. It defines integral and fractional smallness with budget 1/2 (fractional weights in [0,1], including the empty set), qq and qfq_f as suprema of feasible p∈[0,1]p\in[0,1], and states qf≤25⋅5124 qq_f\le25\cdot512^4\,q for nonempty, proper, increasing families. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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