Higher random indestructibility of MAD families

Abstract

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $\kappa$ is indestructible by the higher random forcing $\mathbb Q_\kappa$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_\kappa) = \mathfrak b_\kappa = \mathfrak c_\kappa$ implies the existence $\mathbb Q_\kappa$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.