Cross-Intersecting Families of Partial Permutations

Abstract

For positive integers $r$ and $n$ with $r\leq n$, let $\mathcal{P}_{n,r}$ be the family of all sets $\{(x_1,y_1),\dots,(x_r,y_r)\}$ such that $x_1,\dots,x_r$ are distinct elements of $[n]:=\{1,\dots,n\}$ and $y_1,\dots,y_r$ are also distinct elements of $[n]$. $\mathcal{P}_{n,n}$ describes permutations of $[n]$. For $r<n$, $\mathcal{P}_{n,r}$ describes $r$-partial permutations of $[n]$. Families $\mathcal{A}_1,\dots,\mathcal{A}_k$ of sets are said to be cross-intersecting if, for any distinct $i$ and $j$ in $[k]$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. A sharp bound for the sum of sizes of cross-intersecting subfamilies of $\mathcal{P}_{n,n}$ has recently been established by the author. We generalize this bound by showing that, if $\mathcal{A}_1,\dots,\mathcal{A}_k$ are cross-intersecting subfamilies of $\mathcal{P}_{n,r}$, then (i) $\sum_{i=1}^k|\mathcal{A}_i|\leq{n\choose r}\frac{n!}{(n-r)!}$ if $k\leq\frac{n^2}{r}$ and (ii) $\sum_{i=1}^k|\mathcal{A}_i|\leq k{n-1\choose r-1}\frac{(n-1)!}{(n-r)!}$ if $k\geq\frac{n^2}{r}$. We also determine the structures for which the bound is attained when $r<n$. Our main tool is an extension of Katona's cyclic permutation method.