# Higher random indestructibility of MAD families

## TL;DR

This paper characterizes when maximal almost disjoint families at a weakly compact cardinal are indestructible under higher random forcing, linking combinatorial properties to forcing indestructibility and set-theoretic invariants.

## Contribution

It provides a combinatorial criterion for indestructibility of MAD families under higher random forcing at weakly compact cardinals, extending classical results.

## Key findings

- Characterization of indestructibility using combinatorial properties
- Equivalence of certain set-theoretic invariants in this context
- Existence of indestructible MAD families under specific conditions

## 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.