1 · The essentials
These manuscripts connect two ways of estimating when a randomly selected set is likely to have a desired property. One uses whole choices in a covering calculation, while the other allows fractional weights. A Lean-formalized result says that their estimates differ by at most a fixed factor that does not depend on the particular problem, using the same allowance for covering. A second Lean-formalized result establishes a related statement about combining sets and covering what those combinations miss. As an application, the graph manuscript shows that every graph’s edges can be divided into a bounded number of pieces, each easier to find in a random graph in a precisely controlled sense. The pieces are chosen before the random graph is sampled. Finding all the pieces separately does not necessarily find the whole original graph, because their placements need not match at shared vertices.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine dividing a construction kit into a few smaller assemblies before looking through a randomly stocked parts bin. Each assembly may be achievable on its own, but the separate builds need not fit together into the original model; this captures the important limitation of the graph decomposition.
Why it matters
The threshold comparison establishes that allowing fractional weights changes the estimate by at most a universal factor, without changing the covering budget. The graph application turns that comparison into a decomposition statement with bounds independent of the graph. The requirement that the pieces be fixed before sampling makes the conclusion stronger than one that could choose convenient pieces afterward.
2 · Going further
The threshold problem asks whether integral and fractional expectation thresholds can differ by more than a universal constant factor. The second manuscript establishes that they cannot, with the same covering budget, and is marked as Lean-formalized. The source identifies this as Talagrand’s conjecture. It does not provide the definitions of the two threshold calculations or the details of their comparison argument.
The discrete-convexity statement gives a more explicit covering conclusion. There is a universal integer such that, whenever an arbitrary family has Bernoulli product measure at least , the sets not contained in a union of members of that family admit a cover of total cost at most at the same density. This third manuscript is also marked as Lean-formalized. Its abstract identifies the key structure—combining a bounded number of members of a high-measure family and covering the remaining sets—but does not describe the proof mechanism or the precise cost formula.
The graph manuscript presents an application: every graph has an edge partition into a universally bounded number of fixed pieces, each with ordinary containment threshold at most a universal constant times the original graph’s integral expectation threshold. The manuscript shows the conjecture attributed to Ascoli, He, Park, and Talagrand. The partition must be selected before the random host is sampled, but the embeddings of different pieces need not agree on vertices they shared in the original graph. The repository presents the threshold and discrete-convexity results alongside this application; the supplied abstracts do not spell out the intermediate argument connecting them.
Useful notions
- Sets, families of sets, and unions
- Independent random selection
- Coverings and fractional weights
- Graphs, edges, and embeddings
- Thresholds for random containment
What was known before / what is new
Before
The source identifies the equivalence of integral and fractional expectation thresholds as a conjecture of Talagrand. It also names Talagrand’s discrete-convexity conjecture and the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. No earlier bounds or chronology are supplied.
New
The threshold manuscript establishes a universal-factor comparison with the same covering budget, and the discrete-convexity manuscript establishes the stated high-measure covering conclusion. Both are marked as Lean-formalized. The graph manuscript shows that every graph admits the stated fixed-piece decomposition, with universal bounds on the number of pieces and their containment thresholds.
Limits and assumptions
The discrete-convexity conclusion requires Bernoulli product measure at least and keeps the covering density unchanged. The threshold comparison retains the same covering budget. The graph statement controls separate containment of fixed edge pieces, not compatible embeddings that reconstruct the original graph. Lean formalization is explicitly identified for the threshold-equivalence and discrete-convexity manuscripts, not for the graph-decomposition manuscript. The supplied text gives neither numerical values for the universal constants nor detailed proof arguments, and it specifies no remaining open problem.
Mini glossary
- Universal constant —
- A fixed bound that does not depend on the particular family or graph under consideration.
- Integral and fractional expectation thresholds —
- The two covering-based threshold estimates compared in the source, distinguished by whole choices versus fractional weights.
- Bernoulli product measure —
- A probability model in which elements are selected independently at a specified density.
- Edge partition —
- A division of a graph’s edges into pieces, with every edge assigned to exactly one piece.
- Embedding —
- A placement of a graph or one of its pieces inside a host graph that preserves its edges.
- Containment threshold —
- A threshold describing when a random host is likely to contain the specified graph or piece.
Keywords : Expectation thresholds · Discrete convexity · Graph decompositions
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Graph Decompositions at the Integral Expectation Threshold
05/10/2026We 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.
Integral and fractional expectation thresholds are equivalent
23/09/2026We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.
Talagrand’s discrete-convexity conjecture
23/09/2026We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least , the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.