Result 175, Combinatorics

Talagrand’s expectation thresholds, discrete convexity, and graph decompositions

Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.

Lean formalization Proof

The bigger picture

Why it matters

Two ways of estimating when a random structure should exhibit a desired property may look different: one uses whole choices, the other allows fractional weights. The manuscripts report that their estimates differ only by a universal factor.

What changes?

The manuscripts compare integral expectation thresholds, density benchmarks built from covers using whole choices, with fractional versions allowing weighted choices. They report equivalence within a universal constant factor, with the same covering budget. They also report that every graph's edges partition into a universally bounded number of fixed pieces. Each piece's ordinary containment threshold, the random-host density needed for its appearance, is at most a universal constant times the original graph's integral expectation threshold. The partition precedes sampling.

What does that help mathematicians do?

The discrete-convexity claim supplies a precise covering consequence. For a universal integer k, whenever an arbitrary family has Bernoulli product measure at least one minus one divided by k, the sets not contained in a union of k family members admit a cover of total cost at most one-half, at the same density. Here elements are selected independently at that density. This turns high probability into a bounded-cost description of failures of containment.

Are there practical applications?

The immediate value is foundational: fractional covering arguments can estimate integral expectation thresholds without losing more than a universal factor. The graph result identifies a bounded collection of patterns individually appearing at a controlled density. It does not establish compatible assembly of the original graph: separate embeddings need not agree on shared vertices.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Graph Decompositions at the Integral Expectation Threshold

October 5, 2026 11 pages

We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. Every graph admits a partition into a universally bounded number of fixed edge pieces, each having ordinary containment threshold at most a universal constant times the original graph's integral expectation threshold. The partition is chosen before sampling the random host, and the separate embeddings of the pieces need not agree on shared vertices.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026,
  author = {{OpenAI}},
  title = {{Graph Decompositions at the Integral Expectation Threshold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf}{OAI:Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026}},
  year = {2026}
}

Integral and fractional expectation thresholds are equivalent

September 23, 2026 12 pages Main result formalized in Lean

We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026,
  author = {{OpenAI}},
  title = {{Integral and fractional expectation thresholds are equivalent}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf}{OAI:Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026}},
  year = {2026}
}

Talagrand’s discrete-convexity conjecture

September 23, 2026 10 pages Main result formalized in Lean

We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least $1-1/k$ , the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Talagrands-discrete-convexity-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{Talagrand's discrete-convexity conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf}{OAI:Talagrands-discrete-convexity-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/175.md.

Talagrand’s expectation thresholds, discrete convexity, and graph decompositions

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Talagrand's expectation-threshold conjecture asks whether the fractional and integral expectation thresholds are comparable by an absolute constant. The formalized result proves $q_f(\mathcal F)\le25\cdot512^4 q(\mathcal F)$ for every nonempty proper increasing family on a finite nonempty ground set. Both thresholds use cover budget $1/2$ , and fractional covers may assign weights to every subset, including the empty set.

The formalized result proves Talagrand's discrete-convexity assertion with $k=2^{75}$ . If a family of subsets of a finite nonempty ground set has Bernoulli- $p$ measure at least $1-1/k$ , then the sets not contained in a union of $k$ members form a $p$ -small family: they have a containment cover of total $p$ -cost at most $1/2$ . This holds for every $0<p<1$ , with repeated members allowed in the union and no monotonicity assumption.

Comparator links

Result	Comparator statement
Integral–fractional expectation-threshold comparison	TalagrandExpectationThreshold.lean
Talagrand discrete convexity	TalagrandDiscreteConvexity.lean