Talagrand's discrete convexity conjecture

Let $\mu_p$ be the product Bernoulli measure on subsets of $[N]$ . A family $\mathcal A$ is $p$ -small if there are sets $I$ with $\sum p^{|I|}\le1/2$ such that every member of $\mathcal A$ contains one of them. For a family $\mathcal D$ and an integer $k$ , let $E_k(\mathcal D)$ be the sets contained in no union of $k$ members of $\mathcal D$ . By analogy with his Gaussian convexity question, Talagrand (2010, Conjecture 7.1; also Research Problem 13.3.2 in his 2021 book) conjectured that a fixed number of members of a sufficiently likely family covers everything outside a $p$ -small family. Is there a universal integer $k$ such that $\mu_p(\mathcal D)\ge1-1/k$ implies that $E_k(\mathcal D)$ is $p$ -small, for every $N$ , every $p\in(0,1)$ and every family $\mathcal D$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Probabilistic combinatorics; Bernoulli product measures and covers
Posed by: Michel Talagrand (Conjecture 7.1, STOC 2010; Research Problem 13.3.2, Upper and Lower Bounds for Stochastic Processes, 2nd ed., 2021)
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: 36 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: with $k=2^{75}$ , for every $N\ge1$ , $p\in(0,1)$ and $\mathcal D\subseteq2^{[N]}$ , $\mu_p(\mathcal D)\ge1-1/k$ implies $E_k(\mathcal D)$ is $p$ -small. Consequences: a containment cover for large values of positive selector processes (recovering Park-Pham's theorem), and with Li's fractional result and the companion rounding theorem, $E_2(\mathcal D)$ is $(p/C)$ -small when $\mu_p(\mathcal D)>1/2$ . Earlier, Hua-Song-Tudose derived a discrete statement with three unions at density $p^C$ and Park one with two unions at a $\log\log(1/p)$ loss. The constant $k$ is far from optimal and the Gaussian convexity question is a different problem.

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. Its main theorem is the essential input of the family's graph-decomposition manuscript (October 5, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Conjecture 7.1 as the manuscript states it, at the original density $p$ (no loss in $p$ ), with $k=2^{75}$ and no monotonicity on $\mathcal D$ . lean/formalization.yaml lists TalagrandDiscreteConvexity.json (declaration OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity, file OAI/Combinatorics/DiscreteConvexity/Main.lean). The statement TalagrandDiscreteConvexity.lean was read here; not rebuilt here. For every $N\ge1$ , $0<p<1$ and family $D$ of subsets of Fin N with Bernoulli mass at least $1-2^{-75}$ , the family of sets in no union of $2^{75}$ members (repeats allowed) has a cover of cost at most 1/2 at density $p$ . This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.

Talagrand's discrete convexity conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion