Covering of triples by quadruples in bipartite and tripartite settings

Let $A$ and $B$ be disjoint sets of sizes $a$ and $b$, respectively. Let $f(a,b)$ denote the minimum number of quadruples needed to cover all triples $T\subseteq A\cup B$ such that $\mathrm{\mid }T\cap A\mathrm{\mid }\ge 2$. We prove upper and lower bounds on $f(a,b)$ and use them to derive upper bounds for the $(n,4,3,4)$-lottery problem.