Cross-intersecting families and primitivity of symmetric systems

Abstract

Let $X$ be a finite set and $\mathfrak{p}\subseteq 2^X$, the power set of $X$, satisfying three conditions: (a) $\mathfrak{p}$ is an ideal in $2^X$, that is, if $A\in \mathfrak{p}$ and $B\subset A$, then $B\in \mathfrak{p}$; (b) For $A\in 2^X$ with $|A|\ge 2$, $A\in \mathfrak{p}$ if $\{x,y\}\in \mathfrak{p}$ for any $x,y\in A$ with $x\ne y$; (c) $\{x\}\in \mathfrak{p}$ for every $x\in X$. The pair $(X,\mathfrak{p})$ is called a symmetric system if there is a group $\Gamma$ transitively acting on $X$ and preserving the ideal $\mathfrak{p}$. A family $\{A_1,A_2,\dots ,A_m\}\subseteq 2^X$ is said to be a cross-$\mathfrak{p}$-family of $X$ if $\{a,b\}\in \mathfrak{p}$ for any $a\in A_i$ and $b\in A_j$ with $i\ne j$. We prove that if $(X,\mathfrak{p})$ is a symmetric system and $\{A_1,A_2,\dots ,A_m\}\subseteq 2^X$ is a cross-$\mathfrak{p}$-family of $X$, then \[\sum_{i=1}^m|{A}_i|\leq\left\{ \begin{array}{cl} |X| & \hbox{if $m\le \frac{|X|}{\alpha (X,\mathfrak{p})}$,} \\ m\, α(X,\, \mathfrak p) & \hbox{if $m\ge \frac{|X|}{\alpha (X,\mathfrak{p})}$,} \end{array}\right.\] where $\alpha (X,\mathfrak{p})=\max \{|A|:A\in \mathfrak{p}\}$. This generalizes Hilton's theorem on cross-intersecting families of finite sets, and provides analogs for cross-$t$-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families.

15 pages

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