Multipartite Hypergraphs Achieving Equality in Ryser’s Conjecture

Abstract

A famous conjecture of Ryser (1967) 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 $$r\le 5$$ , 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.