What was already known
Kahn and Kalai conjectured that the expectation obstruction determines graph-containment thresholds within a logarithmic factor uniformly over growing target graphs. The manuscript distinguishes this graph-specific question from the previously resolved abstract threshold conjecture.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Resolves the second Kahn–Kalai conjecture with a logarithm depending on target edge count.
Limitations
The claim concerns ordinary, not induced, copies of H; p_E uses the manuscript’s one-half expected-copy convention.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- 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_E(n,H)(1+log₂ h) for a universal C.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision