Sperner families of bounded VC-dimension (Q1377738)

Let \({\mathcal F}\) be a Sperner family of subsets of an \(m\)-element set. Assume that there is no \(k\)-element subset \(A\) of the ground set such that \(\{A\cap F:F\in {\mathcal F}\}\) is the power set of \(A\). Frankl conjectured that \(|{\mathcal F}|\leq {m \choose k-1}\) for \(m>2k-4\). The extremal families for \(k=2,3\) are given, and this is used to establish the conjecture for \(k=4\). Let \(f(m,k,l)\) be the maximum of \(|{\mathcal F}|\) if the Sperner condition is relaxed to: there are no \(A_1\subset A_2 \subset \cdots \subset A_{l+1}\) in \({\mathcal F}\). Then \(f(m,k,k-1)= {m \choose k-1}+\cdots+{m\choose 1}\) and it is shown that for \(f(m,k,k)\) there are several extremal families.