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

Given integers r > 2$r\geqslant 2$ and n,t > 1$n,t\geqslant 1$ we call families F1,MIDLINE HORIZONTAL ELLIPSIS,Fr subset of P([n])$\mathcal {F}_1,\dots ,\mathcal {F}_r\subseteq \mathcal {P}([n])$ r$r$-cross t$t$-intersecting if for all Fi is an element of Fi$F_i\in \mathcal {F}_i$, i is an element of[r]$i\in [r]$, we have |boolean AND i is an element of[r]Fi|> t$\vert \bigcap _{i\in [r]}F_i\vert \geqslant t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross-intersecting families. In particular, we determine the maximum of n-ary sumation j is an element of[r]|Fj|$\sum _{j\in [r]}\vert \mathcal {F}_j\vert$ for r$r$-cross t$t$-intersecting families in the cases when these are k$k$-uniform families or arbitrary subfamilies of P([n])$\mathcal {P}([n])$. Only some special cases of these results had been proved before. We obtain the aforementioned theorems as instances of a more general result that considers measures of r$r$-cross t$t$-intersecting families. This also provides the maximum of n-ary sumation j is an element of[r]|Fj|$\sum _{j\in [r]}\vert \mathcal {F}_j\vert$ for families of possibly mixed uniformities k1, horizontal ellipsis ,kr$k_1,\ldots ,k_r$.