TUHH / Institut für Mathematik / Forschungsgebiete / r-cross t-intersecting families via necessary intersection points

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

## Description

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. 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 or arbitrary subfamilies of $$\mathscr{P}([n])$$. 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 $$r$$-cross $$t$$-intersecting families. This also provides the maximum of $$\sum_{j\in [r]}\vert\mathcal{F}_j\vert$$ for families of possibly mixed uniformities $$k_1,\ldots,k_r$$.