A family of extremal hypergraphs for Ryser's conjecture

TL;DR

This paper introduces a new infinite family of extremal hypergraphs related to Ryser's Conjecture, expanding known cases and demonstrating the conjecture's complexity through a large variety of non-isomorphic examples.

Contribution

The authors construct a new infinite family of extremal hypergraphs for Ryser's Conjecture based on the existence of projective planes, and define the Ryser poset to analyze their structure.

Findings

Existence of extremal hypergraphs whenever a projective plane of order r-2 exists.

The number of extremal hypergraphs' maximal and minimal elements grows exponentially with rom the structure of the Ryser poset.

The construction broadens the set of known extremal examples, highlighting the conjecture's difficulty.

Abstract

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order $r-1$ exists. We produce a new infinite family of $r$-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order $r-2$ exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the {\em Ryser poset} of extremal intersecting $r$-partite $r$-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in $\sqrt{r}$. This provides further evidence for the difficulty of Ryser's Conjecture.