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.