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 F ⋯ ⊆ P ( [ ] ) $\mathcal {F}_1,\dots ,\mathcal {F}_r\subseteq \mathcal {P}([n])$ $r$ -cross $t$ -intersecting if for all i ∈ $F_i\in {F}_i$ $i\in [r]$ have | ∩ $\vert \bigcap _{i\in [r]}F_i\vert \geqslant t$ . We obtain a strong generalisation of the classic Hilton–Milner theorem on cross-intersecting families. In particular, determine maximum ∑ j $\sum _{j\in [r]}\vert {F}_j\vert$ in cases when these are k $k$ -uniform or arbitrary subfamilies Only some special results had been proved before. aforementioned theorems as instances more general result that considers measures This also provides possibly mixed uniformities … $k_1,\ldots ,k_r$