Multipartite hypergraphs achieving equality in Ryser's conjecture

A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We also improve the general lower bound on $f(r)$, showing that $f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the $r=3$ case fails for all $r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an $r$-partite hypergraph there exists a set $S$ of size at most $r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.

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