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

Abstract

Given integers $r\ge 2$ and $n,t\ge 1$ we call families ${\mathcal{F}}_1,\dots ,{\mathcal{F}}_r\subseteq \mathcal{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in {\mathcal{F}}_i$, $i\in [r]$, we have $|{\bigcap }_{i\in [r]}{F}_i|\ge t$. We obtain a strong generalisation of the classic Hilton-Milner theorem on cross intersecting families. In particular, we determine the maximum of ${\sum }_{j\in [r]}|{\mathcal{F}}_j|$ for $r$-cross $t$-intersecting families in the cases when these are $k$-uniform families or arbitrary subfamilies of $\mathcal{P}([n])$. 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 $r$-cross $t$-intersecting families. This also provides the maximum of ${\sum }_{j\in [r]}|{\mathcal{F}}_j|$ for families of possibly mixed uniformities $k_1,\dots ,k_r$.

13 pages