Families of finite sets with three intersections (Q761458)

Let \(0\leq p_ 1<p_ 2<...<p_ s<k<n\) be integers, and X a finite set of cardinality n. A family \({\mathcal F}\) of k-subsets of X is an \((n,k,\{p_ 1,...,p_ s\})\)-system if for every \(F_ 1,F_ 2\in {\mathcal F}\), \(F_ 1\neq F_ 2\), we have \(| F_ 1\cap F_ 2| \in | \{p_ 1,...,p_ s\}.\) Let \(m(n,k,\{p_ 1,...,p_ s\})\) denote the maximum cardinality of an \((n,k,\{p_ 1,...,p_ s\})\)-system. In this paper, necessary and sufficient conditions for \(m(n,k,\{0,a,b\})=O(n)\) and \(m(n,k,\{0,a,b\}\geq O(n^ 2)\) are given.