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

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.