Christian Bey
An intersection theorem for weighted sets
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock
MSC:
05D05 Extremal set theory
Abstract: A weight function $\omega : 2^{[n]}\rightarrow \mathbb{R}_{\ge 0}$
from the set of all subsets of $[n]=\{1,\ldots,n\}$ to the nonnegative real numbers is called
shift--monotone in $\{m+1,\ldots,n\}$ if $\omega (\{i_1,\ldots,i_m\})\ge \omega (\{j_1,\ldots,j_m\})$
holds for all $\{i_1,\ldots,i_m\}$, $\{j_1,\ldots,j_m\} \subseteq [n]$ with $i_\ell \le j_\ell, \ell=1,\ldots,m$, and if $\omega (I)\ge \omega (J)$ holds for all $I, J \subseteq [n]$ with
$I \subseteq J$ and $J\setminus I\subseteq \{m+1,\ldots,n\}$. A family
$\mathcal{F}\subseteq 2^{[n]}$ is called intersecting in $[m]$ if
$F \cap G \cap [m]\not= \emptyset$ for all $F, G \in \mathcal{F}$. Let
$\omega(\mathcal{F})=\sum_{F\in\mathcal{F}} \omega (F)$. We show that
$\max\{\omega(\mathcal{F}):\mathcal{F}\subseteq 2^{[n]}, \mathcal{F} \text{ is intersecting in } [n]\}$
$=$ $\max\{\omega(\mathcal{F}):\mathcal{F}\subseteq 2^{[n]}, \mathcal{F}$ is intersecting in
$[m]\}$ provided that $\omega$ is shift--monotone in $\{m+1,\ldots,n\}$.
An application to the poset of colored subsets of a finite set is given.
Keywords: intersection theorem, intersecting families