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 $H^\bullet \subseteq H$ which is homomorphically equivalent to $H$. We use the vertex transitivity and strong independence number of almost-uniform $3$-qualitative independence hypergraphs, $3\text{-}AUQI(2n+1,2)$, to show that $3\text{-}AUQI(2n+1,2)$ are cores for infinitely many $n$. Consequently, we establish that $3\text{-}QI(9,2)$ is a core and also obtain a lower bound on the strong chromatic number of $3$-qualitative independence hypergraphs $3\text{-}QI(2n+1,2)$.