Nordhaus–Gaddum-type problems for lines in hypergraphs

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 n 2 ; and the minimum value of their sum is between Ω(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.A hypergraph is an ordered pair (V, H) such that V is a set and H ⊆ 2 V is a family of subsets of V ; elements of V are the vertices of the hypergraph and members of H are its hyperedges; a hypergraph is called k-uniform if each of its hyperedges consists of k vertices; i.e. H ⊆ V k . In studying problems related to a De Bruijn-Erdős theorem, Chen and Chvátal [5] defined lines in 3-uniform hypergraphs. Specifically, for any u, v ∈ V , the line uv is defined as uv = {u, v} ∪ {p : {u, v, p} ∈ H} .