Infinitely Many Almost-Uniform 3-Qualitative Independence Hypergraphs are Cores

Qualitative independence is a property associated with the column vectors of well-known combinatorial designs known as covering arrays and qualitative independence hypergraphs characterize the existence of covering arrays on hypergraphs. The core of a hypergraph H is an interesting structural characteristic to study, as it is the unique (up to isomorphism), minimal subhypergraph H^• ⊆ H which is homomorphically equivalent to H. We use the vertex transitivity and strong independence number of almost-uniform $3$-qualitative independence hypergraphs, 3-AUQI(2n+1,2), to show that 3-AUQI(2n+1,2) are cores for infinitely many n. Consequently, we establish that 3-QI(9,2) is a core and also obtain a lower bound on the strong chromatic number of $3$-qualitative independence hypergraphs 3-QI(2n+1,2).