Christian Bey, Konrad Engel
Old and new results for the weighted $t$--intersection problem via AK--methods
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

05D05 Extremal set theory

06A07 Combinatorics of partially ordered sets

Abstract: Let $[n]:=\{1,\ldots,n\}$, $2^{[n]}$ be the power set of $[n]$ and $s\in [n]$. A family $\mathcal{F}\subseteq 2^{[n]}$
is called {\em $t$-intersecting in $[s]$} if
\[
|X_1\cap X_2\cap [s]|\ge t \text{ for all } X_1, X_2\in\mathcal{F} .
\]
Let $\omega:2^{[n]}\rightarrow \mathbb{R}_{+}$ be a given weight function and
\[
M_s(n,t;\omega):=\max\{\omega(\mathcal{F}) : \mathcal{F} \text{ is $t$-intersecting in $[s]$}\} .
\]
For several weight functions, the numbers $M_n(n,t;\omega)$ can be determined using three important methods
of Ahlswede and Khachatrian: Generating Sets \cite{ahlk97}, Comparison Lemma \cite{ahlk98}, and
Pushing--Pulling \cite{ahlk97a}. We survey these methods.

Also, sufficient conditions on $\o$ for the equality
\[
M_s(n,t;\omega)=M_n(n,t;\omega)
\]
are presented which simplify the method of Generating Sets.
In addition, analogous conditions are given for the case that $|\cap_{X\in\mathcal{F}}X| $t$-intersection).

Applications of these methods include new intersection theorems for chain-- and star products.


Keywords: intersection theorem, intersecting family, star product, chain product, AK method