Strong Colouring of the Qualitative Independence Hypergraph $3\text{-}QI(11,2)$

Abstract

We determine the strong independence number of the qualitative independence hypergraph, $3\text{-}QI(11,2)$, using a technique that involves considering its vertices as subsets of $\{1,2,\dots ,11\}$ and assessing them as intersecting set systems. This gives the maximum size of colour classes in any strong colouring and thus, a lower bound on the strong chromatic number of $3\text{-}QI(11,2)$. We leverage this bound along with an upper bound of the strong chromatic number of $3\text{-}QI(10,2)$, to consequently, establish that the covering array number of $3\text{-}QI(11,2)$, $CAN(3\text{-}QI(11,2),2)=11$ and give a sufficient condition for a hypergraph $H$ to have $CAN(H,2)=11$.