OpenAI · 162-01 · 2026-09-27 · Combinatorics
Balanced counterexamples to Ryser's conjecture at prime orders
preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026
Family 162. Counterexamples to Ryser's covering and Gyárfás's tree-cover conjectures.
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
Disproves Ryser's covering conjecture by constructing intersecting -partite, -uniform hypergraphs with covering number , rather than the predicted bound , for every sufficiently large prime . A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.
For every sufficiently large prime q, we construct a finite intersecting -partite -uniform hypergraph with covering number and exactly nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.
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 162: Counterexamples to Ryser’s covering conjecture. 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.