Nordhaus-Gaddum-type problems for lines in hypergraphs

TL;DR

This paper investigates the symmetric problem of counting lines in hypergraphs and their complements, establishing bounds and exact values for various classes, extending Nordhaus-Gaddum-type results.

Contribution

It introduces bounds for the number of lines in hypergraphs and their complements, including exact results for special classes like Euclidean spaces and projective planes.

Findings

Minimum product of lines in hypergraph and complement is inom{n}{2}

Minimum sum of lines is between (n) and (n  log n)

Exact bounds for hypergraphs from Euclidean space, projective plane, and trees

Abstract

We study the number of lines in hypergraphs in a more symmetric setting, where both the hypergraph and its complement are considered. In the general case and in some special cases, the lower bounds on the number of lines are much higher than their counterparts in single hypergraph setting or admit more elegant proofs. We show that the minimum value of product of the number of lines in both hypergraphs on $n$ points is easily determined as $\binom{n}{2}$; and the minimum value of their sum is between $\Omega(n)$ and $O(n \log n)$. We also study some restricted classes of hypergraphs; and determine the tight bounds on the minimum sum when the hypergraph is derived from an Euclidean space, a real projective plane, or a tree.