Arrow relations on families of finite sets (Q1182739)

Let \(n\), \(m\), \(a\), \(b\) be positive integers. Then \((n,m)\to (a,b)\) means: If \(X\) is a set with \(| X|=n\) and if \(\mathfrak F\) is a set of subsets of \(X\) with \(| {\mathfrak F}|\geq m\) then there exists a set \(Y\subseteq X\) with \(| X|=a\) such that \({\mathfrak F}_ Y:=\{F\cap Y\mid F\in {\mathfrak F}\) has at least \(b\) elements. In this paper the author deals with the problem to determine the maximum integer \(m\) such that \((n,m)\to (n-1,m-k)\), where \(n\), \(k\) are given integers with \(n\geq 4\), and \(k\in\{4,5,6\}\). He proves: (1) \((n,m)\to(n-1,m-4)\) for all \(m\leq \lceil 17n/6\rceil\), (2) \((n,m)\to(n-1,m-5)\) for all \(m\leq \lceil 13n/4\rceil\), (3) \((n,m)\to(n-1,m-6)\) for all \(m\leq 7n/2\) for \(n\equiv 0(\bmod 4)\) (resp. \(\leq \lceil 7n/2+1\rceil\) for \(n\not\equiv 0(\bmod 4)\). It is also proved that these bounds are best possible by exception of the case \(n=7\) in (2). These theorems settle the smallest open cases which are not contained in existing theorems.