Lines in hypergraphs

TL;DR

This paper generalizes a classical hypergraph theorem by allowing certain configurations, identifying new extremal families, and extending the understanding of lines in hypergraphs.

Contribution

It extends the De Bruijn-Erdos theorem to hypergraphs with relaxed conditions, revealing additional extremal structures such as complements of Steiner triple systems.

Findings

The number of lines is at least the number of vertices under new conditions.

Extremal cases include near-pencils, finite projective planes, and complements of Steiner triple systems.

The theorem's equivalence to a line-counting property in 3-uniform hypergraphs is established.

Abstract

One of the De Bruijn - Erdos theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvatal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u,v,w} is a hyperedge. With this definition, the De Bruijn - Erdos theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result eneralizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.