Definable discrete sets with large continuum

Abstract

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by iterated Sacks forcing, there is a $\Delta^1_2$ maximal $\mathcal R$-discrete set, and thus the existence of such sets is compatible with the negation of the continuum hypothesis. As an application we find a $\Pi^1_1$ maximal orthogonal family of Borel probability measures in said extension. The basis of this is a new Ramsey theoretic result.