On Intersection Numbers, Total Clique Covers and Regular Graphs

Abstract

Let $\mathcal{A}=\{A_1,\dots ,A_l\}$ be a partition of $[n]$ and $\mathcal{F}=\{S_1,\dots ,S_m\}$ be an intersecting family of distinct nonempty subsets of $[n]$ such that $\mathcal{A}$ and $\mathcal{F}$ are pairwise intersecting families.Then $|\mathcal{F}|\le \frac{1}{2}\prod _{i=1}^l(2^{|A_i|}–2)+\sum _{S⫋[l]}\left(\prod _{i\in S}(2^{|A_i|}–2)\right).$From this result and some properties of intersection graphs on multifamilies, we determine the intersection numbers of $3$, $4$, and $5$-regular graphs and some special graphs.