Size, diversity, minimum degree, sturdiness, dömdödöm
Balázs Patkós
TL;DR
The paper introduces the $(p,q)$-dömdödöm statistic $β_{p,q}({\mathcal{F}})$ for a family of sets, unifying size, minimum degree, diversity, and sturdiness into a single extremal framework. It analyzes the maximum $β_{p,q}(n,k)$ over all $k$-uniform intersecting families, linking its asymptotics to the covering number $τ({\mathcal{F}})$ and a constant $β(q)=β_{0,q}(n,q+1)$, and obtaining exact large-$n$ formulas $β_{p,1}(n,k)=\binom{n-3-p}{k-2-p}$ and $β_{p,2}(n,k)=2\binom{n-5}{k-3-p}-\binom{n-7}{k-5-p}$. The proofs combine maximal intersecting family structure, minimal covers, and covering-number bounds, and they identify extremal constructions such as a Fano-plane-based family. This work consolidates several classical invariants in extremal set theory under a single parameterized framework and highlights the principal open problem of determining $β(q)$.
Abstract
For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-dömdödöm} of a family $\mathcal{F}\subseteq 2^{[n]}$ is $β_{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 $β_{p,q}(n,k)$ of $β_{p,q}(\mathcal{F})$ over all $k$-uniform intersecting families $\mathcal{F}\subset 2^{[n]}$. We determine the order of magnitude of $β_{p,q}(n,k)$ for all fixed $p,q,k$. We relate the asymptotics of $β_{p,q}(n,k)$ to the constant value of $β_{0,q}(n,q+1)$ and establish $β_{p,1}(n,k)=\binom{n-3-p}{k-2-p}$ and $β_{p,2}(n,k)=2\binom{n-5}{k-3-p}-\binom{n-7}{k-5-p}$ if $n$ is large enough.