Size, diversity, minimum degree, sturdiness, d\"omd\"od\"om

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-d\"omd\"od\"om} of a family $\mathcal{F}\subseteq 2^{[n]}$ is $\beta_{p,q}(\mathcal{F})=\min\{|\mathcal{F}(A,\overline{B})|:|A|=p,|B|=q, A\cap B=\emptyset, A,B\subseteq [n]\}$. 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 $\beta_{p,1}(n,k)=\binom{n-3-p}{k-2-p}$ and $\beta_{p,2}(n,k)=2\binom{n-5}{k-3-p}-\binom{n-7}{k-5-p}$ if $n$ is large enough.