The VC-dimension of Sperner systems (Q1296751)

A set system \(\mathcal F\) on the finite underlying set \(X\) shatters a set \(D \subset X\) if the trace of \(\mathcal F\) on \(D\) is the power-set of \(D.\) The maximum size of a set shattered by \(\mathcal F\) is the VC-dimension of the family \(\mathcal F\). P. Frankl started to investigate maximum Sperner families having bounded VC-dimension. This paper evaluates the maximum possible VC-dimension of a Sperner family and gives an upper bound for the size of a Sperner family with this maximum VC-dimension.