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 $|A|\geq 2$, $A\in \mathfrak p$ if $\{ x,y\}\in \mathfrak p$ for any $x,y\in A$ with $x\neq y$; (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 $\{ A_1,A_2,\ldots,A_m\}\subseteq 2^X$ 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 $i\neq j$. We prove that if $(X,\mathfrak p)$ is a symmetric system and $\{ A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is a cross-$\mathfrak{ p}$-family of $X$, then \[\sum_{ i=1}^m|{ A}_i|\leq\left\{ \begin{ array}{ cl} |X| & \hbox{ if $m\leq \frac{ |X|}{ \alpha(X,\, \mathfrak p)}$,} \\ m\, \alpha(X,\, \mathfrak p) & \hbox{ if $m\geq \frac{ |X|}{ \alpha{ (X,\, \mathfrak p)}}$,} \end{ array}\right.\] 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.
Categories: math.CO
