Cross-intersecting families and primitivity of symmetric systems

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${x,y}\in \mathfrak p$forany$x,y\in A$with$x\neq y$;(c)${x}\in \mathfrak p$forevery$x\in X$. The pair $(X,\mathfrak{p})$ is called a symmetric system if there is a group $\mathrm{\Gamma }$ 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$i\neq j$\mathrm{.}Weprovethatif$(X,\mathfrak p)$ is a symmetric system and $\{A_1,A_2,\dots ,A_m\}\subseteq 2^X$ is a cross-$\mathfrak{p}$-family of $X$, then $$\sum _{i=1}^m\mathrm{\mid }{A}_i\mathrm{\mid }\le \{\begin{array}{cl}\mathrm{\mid }X\mathrm{\mid }& \text{if }m\le \frac{\mathrm{\mid }X\mathrm{\mid }}{\alpha (X,\mathfrak{p})}\text{,}\\ m\alpha (X,\mathfrak{p})& \text{if }m\ge \frac{\mathrm{\mid }X\mathrm{\mid }}{\alpha (X,\mathfrak{p})}\text{,}\end{array}$$ 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.