We show that ZF + DC + “all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

We construct a Borel maximal cofinitary group.

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal κ, there are no κ‐mad families.

We show that the non-existence of mad families is equiconsistent with \, answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that \ “there is no maximal independent set in G” is equiconsistent with \ “there exists an inaccessible cardinal”.

Starting from a model with a Laver-indestructible supercompact cardinal $$\kappa $$, we construct a model of $$ZF+DC_{\kappa }$$ where there are no $$\kappa $$ -mad families.