The definability of combinatorial families of reals, such as mad families, has a long history. The constructible universe \(L\) is a good model for definability, for its nice structural properties. On the other hand, as a rule of thumb, the universe can't be too far from \(L\) if it allows for low projective witnesses of such families. Thus it makes sense to look at forcing extensions of \(L\).

We show that after a countable support iteration of Sacks forcing or splitting forcing (or many others) over \(L\), every analytic hypergraph on a Polish space has a \(\mathbf\Delta_2\) maximal independent set. This means that in the models obtained by these iterations, most types of interesting “maximal families” have \(\mathbf\Delta_2\) witnesses. In particular, this solves an open problem of Brendle, Fischer and Khomskii by providing a model with a \(\Pi_1\) mif (maximal independent family) while the independence number \(\mathfraki\) is bigger than \(\aleph_1\).