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

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 $$\mathcal{A}$_1, $\mathcal{A}$_2, …, $\mathcal{A}$_kare said to be \emph{cross-t-intersecting} if for anyi$and$j$in${1, 2, …, k}$with$i \neq j$,anysetin$$\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 $\kappa \le \mathrm{\mid }\mathcal{F}\mathrm{\mid }$ such that for any $k\ge \kappa$, both the sum and the product of sizes of any $k$ cross-$t$-intersecting sub-families ${\mathcal{A}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathcal{A}}_k$ (not necessarily distinct or non-empty) of $\mathcal{F}$ are maxima if ${\mathcal{A}}_1=\mathrm{.}\mathrm{.}\mathrm{.}={\mathcal{A}}_k=\mathcal{L}$ 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.