Size, diversity, minimum degree, sturdiness, dömdödöm

Abstract

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}:A\subseteq F,B\cap F=\varnothing \}$. The \textbf{$(p,q)$-dömdödöm} of a family $\mathcal{F}\subseteq 2^{[n]}$ is ${\beta }_{p,q}(\mathcal{F})=\min \{|\mathcal{F}(A,\overline{B})|:|A|=p,|B|=q,A\cap B=\varnothing ,A,B\subseteq [n]\}$. This definition encompasses size, diversity, minimum degree, and sturdiness as special cases. We investigate the maximum possible value ${\beta }_{p,q}(n,k)$ of ${\beta }_{p,q}(\mathcal{F})$ over all $k$-uniform intersecting families $\mathcal{F}\subset 2^{[n]}$. We determine the order of magnitude of ${\beta }_{p,q}(n,k)$ for all fixed $p,q,k$. We relate the asymptotics of ${\beta }_{p,q}(n,k)$ to the constant value of ${\beta }_{0,q}(n,q+1)$ and establish ${\beta }_{p,1}(n,k)=\left(\genfrac{}{}{0ex}{}{n-3-p}{k-2-p}\right)$ and ${\beta }_{p,2}(n,k)=2\left(\genfrac{}{}{0ex}{}{n-5}{k-3-p}\right)-\left(\genfrac{}{}{0ex}{}{n-7}{k-5-p}\right)$ if $n$ is large enough.

6 pages