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Math-2613 · mathematics · Combinatorics

Uniform variance-influence bound for hypergraph properties

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed $r>=3$, does $C_r$ exist such that for every $n>=r$, every p in (0,1), and every Boolean property of r-uniform hypergraphs invariant under vertex relabeling, $Var_p(f)<=C_r$ $I_p(f)/(\log $ $n)^(r/(r-1))$?

Short English statement

For each fixed $r>=3$, does $C_r$ exist such that for every $n>=r$, every p in (0,1), and every Boolean property of r-uniform hypergraphs invariant under vertex relabeling, $Var_p(f)<=C_r$ $I_p(f)/(\log $ $n)^(r/(r-1))$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2613-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2613

For each fixed uniformity $r\ge3$, the selected manuscript asserts the displayed variance-to-influence bound for every $n\ge r$, every bias $p\in(0,1)$, and every vertex-invariant Boolean hypergraph property. It makes no monotonicity assumption, so it covers the entire property class in the question.

Uniform variance-influence bound for hypergraph properties

OpenAI's claim

Uniform hypergraph influence: For fixed uniformity $r\ge3$, the paper claims a variance-to-influence inequality with denominator $(\log n)^{r/(r-1)}$ for all biases and all relabeling-invariant Boolean properties. Increasing properties therefore have the corresponding sharp threshold width.

Mathematical statement of the claimed result

Mathematical claim 1.
For each fixed $r\ge3$, every $n\ge r$, every $0<p<1$, and every vertex-permutation-invariant Boolean property $f$ of an $r$-uniform hypergraph on $[n]$, the manuscript claims \[\operatorname{Var}_p(f)\le\frac{C_r}{(\log n)^{r/(r-1)}}I_p(f),\qquad I_p(f)=\sum_{e\in\binom{[n]}r}\mathbb P_p\left(f(x_{e\leftarrow1})\ne f(x_{e\leftarrow0})\right).\] The product measure is Bernoulli$(p)$ on hyperedges. Influence here contains no additional factor $p(1-p)$, and monotonicity is not required.

Outline of the argument

Uniform hypergraph influence: Restrictions leave every hyperedge touching a random vertex block free, so vertex permutations still create large free-edge orbits. The restricted Fourier estimate controls low degrees uniformly in the bias. An $r$-uniform degree argument lifts this to the original spectrum, while influence handles its high-degree remainder; monotone differentiation supplies the threshold consequence.

Formal verification

Family 186 describes Lean coverage for the graph-property threshold width; the hypergraph refinement is not part of this selection.

Uniform hypergraph influence: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

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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