A comment on Ryser's conjecture for intersecting hypergraphs

TL;DR

This paper investigates the minimal edge count in intersecting hypergraphs with large cover number, providing exact results for small r, an almost optimal construction for r=6, and a lower bound for larger r.

Contribution

It offers new bounds and constructions for intersecting hypergraphs with high cover number, advancing understanding of Ryser's conjecture for small and moderate r.

Findings

Exact minimal edge counts for r ≤ 5.

Almost optimal construction for r=6.

Lower bound of approximately 2.764r edges for large r.

Abstract

Let $\tau(\mathcal{H})$ be the cover number and $\nu(\mathcal{H})$ be the matching number of a hypergraph $\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\mathcal{H}$ satisfies the inequality $\tau(\mathcal{H}) \leq (r-1) \nu (\mathcal{H})$. This conjecture is open for all $r \ge 4$. For intersecting hypergraphs, namely those with $\nu(\mathcal{H})=1$, Ryser's conjecture reduces to $\tau(\mathcal{H}) \leq r-1$. Even this conjecture is extremely difficult and is open for all $ r \ge 6$. For infinitely many $r$ there are examples of intersecting $r$-partite hypergraphs with $\tau(\mathcal{H})=r-1$, demonstrating the tightness of the conjecture for such $r$. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting $r$-partite hypergraph be, given that its cover number is as large as possible, namely $\tau(\mathcal{H}) \ge r-1$? In this paper we solve this question for $r \le 5$, give an almost optimal construction for $r=6$, prove that any $r$-partite intersecting hypergraph with $\tau(H) \ge r-1$ must have at least $(3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))$ edges, and conjecture that there exist constructions with $\Theta(r)$ edges.