On the maximum number of points in a maximal intersecting family of finite sets

TL;DR

This paper investigates the maximum size of point sets in maximal intersecting families of finite sets, improving bounds on this maximum and narrowing the gap between known lower and upper estimates.

Contribution

The paper refines the asymptotic upper bound for the maximum number of points in such families, reducing it from four times to three times the known lower bound.

Findings

Asymptotic upper bound is at most three times the lower bound.

Improves previous bounds from four times to three times the lower bound.

Conjectures the explicit upper bound may be only double the lower bound.

Abstract

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the problem of determining or estimating the largest number $N(k)$ of the points in such a family. They also proved by means of an example that $N(k)\geq2k-2+\frac{1}{2}\binom{2k-2}{k-1}$. Much later, Zsolt Tuza proved that the bound is best possible up to a multiplicative constant by showing that asymptotically $N(k)$ is at most $4$ times this lower bound. In this paper we reduce the gap between the lower and upper bound by showing that asymptotically $N(k)$ is at most $3$ times the Erd\H{o}s-Lov\'{a}sz lower bound. Conjecturally, the explicit upper bound obtained in this paper is only double the lower bound.