OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
For every sufficiently large prime s\equiv2\pmod3 and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting (s^n+1)-partite (s^n+1)-uniform hypergraph with covering number s^n+1. This disproves Ryser's covering conjecture in its intersecting case.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
Explore the AGI Clock ↗