OpenAI · 176-01 · 2026-09-24 · Combinatorics
The second Kahn–Kalai conjecture
preprints/The-second-Kahn-Kalai-conjecture-September-24-2026
Family 176. The second Kahn–Kalai conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the second Kahn–Kalai conjecture: for every finite simple graph with edges and at most vertices, its appearance threshold in is at most , with universal . Here is the least density at which every subgraph of has expected copy count at least .
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 to contain an ordinary copy of H is at most , where C is universal. Here is the least density at which every subgraph of H has expected copy count at least one half.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
This preprint is named in the Lean scope note for 176: The second Kahn–Kalai conjecture with an edge-count bound. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.