Intersecting and 2‐intersecting hypergraphs with maximal covering number: The Erdős–Lovász theme revisited.

Erdős and Lovász noticed that an r‐uniform intersecting hypergraph H with maximal covering number, that is, τ(H)=r, must have at least 83r−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 three examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erdős and Lovász, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for r∈{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, τ(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 λ=2. We determined that only three biplanes of the 18 known examples are extremal. [ABSTRACT FROM AUTHOR]