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

TL;DR

This paper introduces the df6md6df6m parameter, unifying various set family properties, and determines its maximum value for intersecting families, revealing asymptotic behaviors and exact formulas for specific cases.

Contribution

It defines a new parameter df6md6df6m that generalizes size, diversity, and other properties, and provides asymptotic and exact results for its maximum in intersecting families.

Findings

Determined the order of magnitude of df6md6df6m for fixed parameters.

Established exact formulas for df6md6df6m when q=1 and q=2.

Connected asymptotics of df6md6df6m to specific base cases.

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{$(p,q)$-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.