On the ratio of maximum and minimum degree in maximal intersecting families

TL;DR

This paper investigates the ratio of maximum to minimum degree in maximal intersecting families of sets, establishing bounds and using projective plane theory to understand their structure and extremal properties.

Contribution

It determines the order of magnitude of the minimal ratio in such families and provides bounds on the maximal ratio, introducing new extremal constructions.

Findings

Established the order of magnitude of the minimum ratio m(n,r).

Provided bounds on the maximum ratio M(n,r).

Used projective plane blocking set theorems for constructions.

Abstract

To study how balanced or unbalanced a maximal intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is we consider the ratio $\mathcal{R}(\mathcal{F})=\frac{\Delta(\mathcal{F})}{\delta(\mathcal{F})}$ of its maximum and minimum degree. We determine the order of magnitude of the function $m(n,r)$, the minimum possible value of $\mathcal{R}(\mathcal{F})$, and establish some lower and upper bounds on the function $M(n,r)$, the maximum possible value of $\mathcal{R}(\mathcal{F})$. To obtain constructions that show the bounds on $m(n,r)$ we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes.