Qualitative independence and Sperner problems for directed graphs
Two \(k\)-partitions, \({\mathcal P}=\{P_ 1,\dots,P_ k\}\) and \({\mathcal P}'=\{P_ 1',\dots,P_ k'\}\) are called qualitatively independent, if \(P_ i\cap P_ j'\) is never empty. Let \(N(n,k)\) denote the largest size of a family of pairwise qualitative independent \(k\)-partitions on \(n\) elements. It is shown that \(\limsup_ n{1\over n}\log N(n,3)\geq.483\) and improved lower bounds are given also for \(4\leq k\leq 13\). The problem is related to that of the Shannon capacity of a graph.
scientific article; zbMATH DE number 96571