Result 176, Combinatorics

The second Kahn–Kalai conjecture with an edge-count bound

Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in $G(n,p)$ is at most $C p_{\mathrm E}(n,H)(1+\log_2 h)$ , with universal C. Here $p_{\mathrm E}$ is the least density at which every subgraph of H has expected copy count at least 1/2.

Lean formalization Proof

The bigger picture

Why it matters

How dense must a random graph be before it contains a prescribed pattern? The manuscript claims that expected counts of the pattern's subgraphs give an upper bound on this transition, with only a logarithmic overhead in the pattern's edge count.

What changes?

For every finite simple graph H with h edges, where h is at least one, and at most n vertices, the manuscript bounds its appearance threshold in G(n,p), the n-vertex graph whose edges occur independently with probability p. The bound is C times the expectation density times (1 + log base 2 of h), with universal C. The expectation density is the least p at which every subgraph of H has expected copy count at least one half.

What does that help mathematicians do?

The result concerns ordinary copies: extra edges between the chosen vertices are allowed. Checking every subgraph accounts for scarce pieces that could obstruct appearance. The claimed bound says that, once all these expected counts reach one half, the appearance threshold is at most a logarithmic factor higher. Researchers can therefore bound thresholds through subgraph counts, with no additional factor depending on n.

Are there practical applications?

Its immediate value is foundational: it connects expected counts of structures with their appearance thresholds in random graphs. For researchers studying particular graph families, the bound provides a route from counting subgraphs to controlling when a whole pattern appears. It does not itself provide an algorithm for finding that pattern.

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.

Manuscript

The second Kahn–Kalai conjecture

September 24, 2026 12 pages Main result formalized in Lean

We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for $G(n,p)$ to contain an ordinary copy of H is at most $C p_{\mathrm E}(n,H)(1+\log_2 h)$ , where C is universal. Here $p_{\mathrm E}(n,H)$ is the least density at which every subgraph of H has expected copy count at least one half.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-second-Kahn-Kalai-conjecture-September-24-2026,
  author = {{OpenAI}},
  title = {{The second Kahn--Kalai conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf}{OAI:The-second-Kahn-Kalai-conjecture-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/176.md.

The second Kahn–Kalai conjecture with an edge-count bound

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

The second Kahn–Kalai conjecture

Scope

The second Kahn–Kalai conjecture compares the threshold for a random graph to contain a fixed graph with its expectation threshold. The formalized result proves the comparison up to a universal factor times $1+\log_2|E(H)|$ , and hence up to a universal factor times $\log_2 n$ , for every graph $H$ with at least one edge and at most $n$ vertices, $n\ge2$ . The bounds use the actual containment and expectation thresholds.

Comparator links

Result	Comparator statement
Second Kahn–Kalai threshold bounds	SecondKahnKalai.lean

At a glance

Subject: Combinatorics
Manuscripts: 1
Pages: 12
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Second Kahn–Kalai conjectureProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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