A Turan Problem for Cartesian Products of Hypergraphs

Let \(S\) and \(T\) be sets with \(|S| = m\) and \(|T| = n\). Let \(S_3, S_2\) and \(T_3, T_2\) be the sets of all \(3\)-subsets (\(2\)-subsets) of \(S\) and \(T\), respectively. Define \(Q((m, 2, 3), (n, 2, 3))\) as the smallest subset of \(S_2 \times T_2\) needed to cover all elements of \(S_3 \times T_3\). A more general version of this problem is initially defined, but the bulk of the investigation is devoted to studying this number. Its property as a lower bound for a planar crossing number is the reason for this focus.