Restricted \(t\)-wise \(\mathcal{L}\)-intersecting families on set systems
Let \(L\) be any ordered set of \(s\) nonnegative integers, \(\lambda=\min L\) and \(t\geq2\). A family \(F\) of subsets of \([n]\) is \textit{\(t\)-wise \(L\)-intersecting} if the size of the intersection of any \(t\) distinct members of \(F\) belongs to \(L\). The authors show that {\parindent=5mm \begin{itemize}\item[1)]if \(t\geq 3\) and \(F\) is a \(t\)-wise \(L\)-intersecting family of subsets from \([n]\) then either \[ \left|\bigcap_{A\in F}A\right|< \lambda\text{ and }|F|=O(n^s) \] or \[ \left|\bigcap_{A\in F}A\right|\geq\lambda\text{ and }|F|\leq \frac{k+s-1}{s+1}\binom{n-\lambda}{s}+\sum_{j=0}^{s-1}\binom{n-\lambda}{j} \] \item[2)]if \(F\) is a \(k\)-uniform \(t\)-wise \(\{\lambda\}\)-intersecting family of subsets from \([n]\) then \(|F|\leq\frac{(n-\lambda)(t-1)}{k-\lambda}\) for all \(n>\lambda+k+\frac{k(k-2)}{t-1}\). \end{itemize}}
scientific article; zbMATH DE number 5860467