Locally thin set families (Q2711614)

A family \(\mathcal A\) of subsets of a finite set \(N\) is called \(k\)-locally thin if for any \(k\) distinct subsets of \(\mathcal A\) there exists an element of \(N\) that is contained in exactly one of them. For \(n=|N|\), let \(M(n,k)\) denote the maximum cardinality of a \(k\)-locally thin family. The main result is a complicated-looking upper bound for \(t(k)=\limsup_{n\to\infty} {1\over n}\log_2 M(n,k)\) for even \(k\). This bound, in particular, implies \(t(k) < 2/k\) for even \(k>2\).