The maximum sum and maximum product of sizes of cross-intersecting families

Abstract

We say that a set $A$ \emph{$t$-intersects} a set $B$ if $A$ and $B$ have at least $t$ common elements. A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if each set in $\mathcal{A}$ $t$-intersects any other set in $\mathcal{A}$. Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ are said to be \emph{cross-$t$-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ $t$-intersects any set in $\mathcal{A}_j$. We prove that for any finite family $\mathcal{F}$ that has at least one set of size at least $t$, there exists an integer $\kappa \leq |\mathcal{F}|$ such that for any $k \geq \kappa$, both the sum and the product of sizes of any $k$ cross-$t$-intersecting sub-families $\mathcal{A}_1, ..., \mathcal{A}_k$ (not necessarily distinct or non-empty) of $\mathcal{F}$ are maxima if $\mathcal{A}_1 = ... = \mathcal{A}_k = \mathcal{L}$ for some largest $t$-intersecting sub-family $\mathcal{L}$ of $\mathcal{F}$. We then study the smallest possible value of $\kappa$ and investigate the case $k < \kappa$; this includes a cross-intersection result for straight lines that demonstrates that it is possible to have $\mathcal{F}$ and $\kappa$ such that for any $k < \kappa$, the configuration $\mathcal{A}_1 = ... = \mathcal{A}_k = \mathcal{L}$ is neither optimal for the sum nor optimal for the product. We also outline solutions for various important families $\mathcal{F}$, and we provide solutions for the case when $\mathcal{F}$ is a power set.