Combining the theorems of Tur\'an and de Bruijn-Erd\H os

TL;DR

This paper generalizes classical theorems in combinatorics by establishing a lower bound on the number of lines in a linear space with specific point-line incidence properties, extending de Bruijn-Erdős theorem to hypergraphs.

Contribution

It introduces a new bound on lines in linear hypergraphs satisfying certain point-set conditions, generalizing the de Bruijn-Erdős theorem for s ≥ 2.

Findings

Proves a sharp lower bound on the number of lines for large n.

Extends the de Bruijn-Erdős theorem to hypergraphs.

Establishes conditions under which the bound is tight.

Abstract

Fix an integer $s \ge 2$. Let $\mathcal{P}$ be a set of $n$ points and let $\mathcal{L}$ be a set of lines in a linear space such that no line in $\mathcal{L}$ contains more than $(n-1)/(s-1)$ points of $\mathcal{P}$. Suppose that for every $s$-set $S$ in $\mathcal{P}$, there is a pair of points in $S$ that lies in a line from $\mathcal{L}$. We prove that $|\mathcal{L}| \ge (n-1)/(s-1)+s-1$ for $n$ large, and this is sharp when $n-1$ is a multiple of $s-1$. This generalizes the de Bruijn-Erd\H os theorem which is the case $s=2$. Our result is proved in the more general setting of linear hypergraphs.