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Math-2624 · mathematics · Combinatorics

Talagrand discrete convexity conjecture

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does a universal integer $k$ exist such that, for every finite ground set $X$, every $p\in(0,1)$, and every family $D\subseteq 2^X$ with $p$-biased measure at least $1-1/k$, the family \[ E_k(D)=\{S\subseteq X:S\text{ is not contained in any union of }k\text{ members of }D\} \] has a cover $G$ satisfying \[ \sum_{I\in G}p^{|I|}\leq\frac12, \] where a cover means that every $S\in E_k(D)$ contains some $I\in G$?

Short English statement

Is there a universal integer $k$ such that, for every finite $X$, every $p\in(0,1)$, and every $D\subseteq2^X$ of $p$-biased measure at least $1-1/k$, the sets $S\subseteq X$ that are not contained in any union of $k$ members of $D$ have a cover $G$ with $\sum_{I\in G}p^{|I|}\leq1/2$, meaning that each such $S$ contains a member of $G$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2624-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2624

The explicit $k=2^{75}$ theorem supplies the requested containment cover with budget one half for every product-measure parameter and arbitrary family, allowing repeated tuple members. Relabelling handles all finite nonempty ground sets; for the empty ground set the exceptional family is empty.

Talagrand discrete convexity conjecture

OpenAI's claim

Talagrand discrete convexity: The paper claims a universal number of family members whose unions leave only a small containment-cover obstruction, provided the original Bernoulli measure is sufficiently close to one. The cover cost is bounded at the original density.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $k=2^{75}$ and $\mu_p$ be product Bernoulli measure on $2^{[N]}$. For every $N\ge1$, $0<p<1$, and arbitrary family $\mathcal D$, define $E_k(\mathcal D)=\{S:S\nsubseteq D_1\cup\cdots\cup D_k\text{ for every }D_1,\ldots,D_k\in\mathcal D\}$. Repetitions among the $D_i$ are allowed. The theorem claims \[\mu_p(\mathcal D)\ge1-1/k\quad\Longrightarrow\quad\exists\mathcal G:\ E_k(\mathcal D)\subseteq\{S:\exists I\in\mathcal G,\ I\subseteq S\},\quad\sum_{I\in\mathcal G}p^{|I|}\le1/2.\] No increasingness or disjointness hypothesis is imposed on $\mathcal D$.

Outline of the argument

Talagrand discrete convexity: A biased orthogonal expansion produces nonnegative subset weights with a global budget. A signed many-row identity turns failure of a union cover into a large higher-power sum of those weights. Large weights give cheap generators directly, while weighted-degree selection and random unions cover the remaining classes; a final density transfer gives the universal union count.

Formal verification

Family 175 describes Lean coverage for the integral/fractional threshold comparison and the discrete-convexity statement; graph decomposition is a later separate consequence.

Talagrand discrete convexity: The manuscript links TalagrandDiscreteConvexity.lean (declaration OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity), TalagrandExpectationThreshold.lean (declaration OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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