A note on intersecting hypergraphs with large cover number

TL;DR

This paper constructs specific intersecting hypergraphs with large cover numbers, providing insights into longstanding conjectures and showing that certain bounds are nearly optimal for all but finitely many cases.

Contribution

It introduces a new construction of r-partite r-uniform intersecting hypergraphs with high cover number, advancing understanding of Ryser's conjecture.

Findings

Constructed hypergraphs with cover number at least r-4 for most r

Answered a question posed by Abu-Khazneh et al.

Indicated that Ryser's conjecture bounds are nearly tight.

Abstract

We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r-4 for all but finitely many r. This answers a question of Abu-Khazneh, Barat, Pokrovskiy and Szabo, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r.