Ryser's conjecture on covers of r-partite hypergraphs

Ryser's conjecture is false, already for intersecting hypergraphs. September 23: for every sufficiently large prime $s\equiv2\pmod3$ and every sufficiently large odd $n$ (threshold depending on $s$), there is an intersecting $(s^n+1)$-partite $(s^n+1)$-uniform hypergraph with $\tau=s^n+1$, against the conjectured $s^n$. September 27: for every sufficiently large prime $q$, an intersecting $(q+1)$-partite $(q+1)$-uniform hypergraph with $\tau=q+1$ and exactly $q+1$ nonisolated vertices in each part. Not given: an explicit rank or example, or anything for small $r$ (cases $r=4,5$ remain open in general).

Ryser's conjecture is false, already for intersecting hypergraphs. September 23: for every sufficiently large prime $s\equiv2\pmod3$ and every sufficiently large odd $n$ (threshold depending on $s$), there is an intersecting $(s^n+1)$-partite $(s^n+1)$-uniform hypergraph with $\tau=s^n+1$, against the conjectured $s^n$. September 27: for every sufficiently large prime $q$, an intersecting $(q+1)$-partite $(q+1)$-uniform hypergraph with $\tau=q+1$ and exactly $q+1$ nonisolated vertices in each part. Not given: an explicit rank or example, or anything for small $r$ (cases $r=4,5$ remain open in general).