$r$-cross $t$-intersecting families via necessary intersection points

This work is concerned with the study of $r$-cross~$t$-intersecting families. Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross~$t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. Our main result determines the maximum sum of measures of $r$-cross $t$-intersecting families for a fairly general type of measures, including the product measure and the uniform measure. In particular, we determine the maximum of $\sum_{j\in [r]}\vert\mathcal{F}_j\vert$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families, families of possibly mixed uniformities $k_1,\ldots,k_r$, or arbitrary subfamilies of $\mathscr{P}([n])$. This solves some of the main open problems regarding $r$-cross $t$-intersecting families and generalises a result by Frankl and Kupavskii and recent independent work by Frankl and Wong H.W., and Shi, Frankl, and Qian. We further use our proof technique of "necessary intersection points" to solve another problem regarding intersecting families and measures posed by Frankl and Tokushige.