Result 186, Combinatorics

Uniform influence and sharp thresholds for graph and hypergraph properties

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0\lt \varepsilon\lt 1/2$ , every nontrivial increasing relabeling-invariant property crosses from probability ε to $1-\varepsilon$ within width $O((\log n)^{-2})$ for graphs and $O_r((\log n)^{-r/(r-1)})$ for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.

Lean formalization Proof

The bigger picture

Why it matters

In a random graph or hypergraph, each possible edge is included independently with probability p. These unreviewed manuscripts claim that any nontrivial property preserved by adding edges and unchanged by vertex relabeling must become likely within a tightly bounded change in p.

What changes?

For graphs with n at least two vertices, the manuscript bounds the transition width by C log(1/(2 epsilon)) divided by (log n) squared, with universal C. The transition runs from probability epsilon to one minus epsilon, for any epsilon strictly between zero and one-half. For simple r-uniform hypergraphs, with exactly r vertices per edge, fixed r at least three gives width at most a constant times (log n) raised to minus r/(r-1), with the constant depending only on r and fixed epsilon.

What does that help mathematicians do?

For hypergraphs, the manuscript also bounds variance, which measures uncertainty in the property's truth, by total influence times a constant depending only on r, divided by (log n) raised to r/(r-1). Total influence sums the probabilities that changing individual edges changes the answer. This holds for every p strictly between zero and one, even without preservation under adding edges. It therefore rules out relabeling-invariant properties that remain uncertain while being too insensitive to edge changes.

Are there practical applications?

Their immediate value is foundational: they provide universal limits on how gradually random graph and fixed-uniformity hypergraph properties can emerge. Researchers can use these limits to check proposed transition behavior without analyzing each property separately. They do not, by themselves, locate a property's threshold or supply an algorithm for detecting it.

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.

2 manuscripts

A uniform influence bound for hypergraph properties

October 5, 2026 8 pages

For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies $\mathop{\mathrm{Var}}\nolimits _p(f)\le C_r I_p(f)/(\log n)^{r/(r-1)}$ . The constant depends only on r, and the bound holds uniformly for all $0\lt p\lt 1$ without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent $r/(r-1)$ , proving the hypergraph threshold-width conjecture of Friedgut and Kalai.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-uniform-influence-bound-for-hypergraph-properties-October-5-2026,
  author = {{OpenAI}},
  title = {{A uniform influence bound for hypergraph properties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf}{OAI:A-uniform-influence-bound-for-hypergraph-properties-October-5-2026}},
  year = {2026}
}

A Sharp Threshold Bound for Monotone Graph Properties

September 25, 2026 10 pages

We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every $0\lt \varepsilon\lt 1/2$ , the edge probabilities at which its probability equals ε and $1-\varepsilon$ differ by at most $C\log(1/(2\varepsilon))/(\log n)^2$ , for a universal constant C.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026,
  author = {{OpenAI}},
  title = {{A Sharp Threshold Bound for Monotone Graph Properties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf}{OAI:A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/186.md.

Uniform influence and sharp thresholds for graph and hypergraph properties

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

A Sharp Threshold Bound for Monotone Graph Properties

Scope

The Friedgut–Kalai sharp-threshold conjecture concerns how quickly a nontrivial increasing graph property appears in the independent-edge random graph. For every $n\ge2$ , every such property invariant under vertex relabeling, and $0<\varepsilon<1/2$ , the formalization proves that the edge-probability interval between probabilities $\varepsilon$ and $1-\varepsilon$ has width at most $2^{19}\log(1/(2\varepsilon))/(\log n)^2$ . The constant is uniform over the graph property, $n$ , and $\varepsilon$ .

Comparator links

Result	Comparator statement
Sharp threshold width for monotone graph properties	SharpThreshold.lean