Size, diversity, minimum degree, sturdiness, dömdödöm

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\bar{B})=\{F\in \mathcal{F}:A\subseteq F,B\cap F=\mathrm{\varnothing }\}$. The $(p,q)\mathbf{\text{-d}}\ddot{\mathbf{\text{o}}}\mathbf{\text{md}}\ddot{\mathbf{\text{o}}}\mathbf{\text{d}}\ddot{\mathbf{\text{o}}}\mathbf{\text{m}}$ of a family $\mathcal{F}\subseteq 2^{[n]}$ is ${\beta }_{p,q}(\mathcal{F})=\min \{\mathrm{\mid }\mathcal{F}(A,\bar{B})\mathrm{\mid }:\mathrm{\mid }A\mathrm{\mid }=p,\mathrm{\mid }B\mathrm{\mid }=q,A\cap B=\mathrm{\varnothing },A,B\subseteq [n]\}$. 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 ${\beta }_{p,1}(n,k)=\left(\genfrac{}{}{0px}{}{n-3-p}{k-2-p}\right)$ and ${\beta }_{p,2}(n,k)=2\left(\genfrac{}{}{0px}{}{n-5}{k-3-p}\right)-\left(\genfrac{}{}{0px}{}{n-7}{k-5-p}\right)$ if $n$ is large enough.