r-cross t-intersecting families via necessary intersection points
Working Groups: Lehrstuhl Diskrete Mathematik
Collaborators (MAT): Yannick Mogge, M. Sc.
Collaborators (External): Pranshu Gupta, Simón Piga, Bjarne Schülke
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 \(k_1,\ldots,k_r\).
