Multiply-intersecting families (Q1186135)

Let \({\mathcal F}\) be a family of distinct subsets of an \(n\)-element set. It is called \(r\)-wise \(t\)-intersecting if any \(r\) elements of \({\mathcal F}\) have at least \(t\) elements in common. The maximum cardinality of such an \({\mathcal F}\) is at least \(2^{n-t}\) for all \(n\geq t\) (let's fix \(t\) elements and take the family \({\mathcal F}\) of all subsets containing these fixed elements). The central result of this paper is that this lower bound is the exact value iff \(n<r+t\) or \(t\leq 2^ r-r-1\) (with the possible but unlikely exception \((r,t)=(3,4))\). The paper also proves results on dually \(r\)-wise \(t\)-intersecting and \(r\)-cross \(t\)- intersecting families, including bounds for the minimum degrees. This paper is rich on results and uses advanced techniques to prove them.