Intersecting and $2$-intersecting hypergraphs with maximal covering number: the Erd\H{o}s-Lov\'asz theme revisited

TL;DR

This paper investigates the minimal size of intersecting hypergraphs with maximal covering number, providing new exact results for small uniformities, uniqueness of extremal examples, and exploring special cases like projective planes and biplanes.

Contribution

It establishes new minimal edge counts for uniformities 4 and 5, proves uniqueness of extremal hypergraphs for certain cases, and analyzes special geometric configurations.

Findings

Unique extremal hypergraph for uniformity 4

Exact minimal edges for uniformity 5 is 13 with 3 examples

Only 3 of 18 known biplanes are extremal

Abstract

Erd\H{o}s and Lov\'asz noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let $q(r)$ denote the minimum number of edges in an intersecting $r$-uniform hypergraph. It was known that $q(3)=6$ and $q(4)=9$. We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric by any means. For uniformity 5, $q(5)=13$, and we found 3 examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erd\H{o}s and Lov\'asz, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for $r\in \{3,4,5,6\}$. For uniformity 6, there is a unique extremal example. In a related question, we try to find $2$-intersecting $r$-uniform hypergraphs with maximal covering number, that is $\tau(H)=r-1$. An infinite family of examples is to take all possible $r$-sets of a $(2r-2)$-vertex set. There is also a geometric candidate: biplanes. These are symmetric 2-designs with $\lambda=2$. We determined that only 3 biplanes of the 18 known examples are extremal.