Traces of antichains (Q1812791)

Let \({\mathcal F}\) be a family of subsets of a set \(X\). Then \({\mathcal F}\) is called \(k\)-dense if there exists \(Y\subseteq X\), \(| Y|=k\) such that \(|\{F\cap Y:F\in{\mathcal F}\}|=2^ k\). The author discusses problems concerning the maximum cardinality of non-\(k\)-dense families and concentrates mainly on families which are antichains.