Counterexamples to Ryser’s covering conjecture

What OpenAI claims

Disproves Ryser's covering conjecture by constructing intersecting (q+1)-partite, (q+1)-uniform hypergraphs with covering number q+1, rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture. (Lean)

Papers (2)

1. Balanced counterexamples to Ryser's conjecture at prime orders

   27 September 2026 · Preprint, not peer reviewed · PDF

   For every sufficiently large prime q, we construct a finite intersecting (q+1)-partite (q+1)-uniform hypergraph with covering number q+1 and exactly q+1 nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.

2. A counterexample to Ryser's covering conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   For every sufficiently large prime s ≡ 2 (mod 3) and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting (sⁿ+1)-partite (sⁿ+1)-uniform hypergraph with covering number sⁿ+1. This disproves Ryser's covering conjecture in its intersecting case.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.