Families of \(4\)-Sets Without Property \(B\)

A family \(\mathcal{F}\) of finite sets is said to have property \(B\) if there exists a set \(S\) such that \(0 < |{S} \cap F| < |F|\) for all \(F \in \mathcal{F}\). Denote by \(m_N(n)\) the least integer \(m\) for which there exists a family \(\mathcal{F}\) of \(m\) \(n\)-element subsets of a set \(V\) of size \(N\) such that \(\bigcup \mathcal{F} = V\) and which does not have property \(B\). We give constructions which yield upper bounds for \(m_N(4)\) for certain values of \(N\).