Counterexamples to Ryser’s covering conjecture. 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.
released 2026-09-27 | 2 theorems · 10 lemmas · 15 proofs · 21,162 words |
PLAY LEVEL 1 »(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.
released 2026-09-23 | 3 theorems · 11 lemmas · 16 proofs · 17,998 words |
PLAY LEVEL 2 »(pdf)
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.