Definable discrete sets with large continuum

Abstract

Let $\mathcal{R}$ be a $\hat{I}{\pounds }_1^1$ 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 $\hat{I}{”}_2^1$ 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 ${\hat{I}}_1^1$ maximal orthogonal family of Borel probability measures in said extension. The basis of this is a new Ramsey theoretic result.

Includes a new result (Proposition 4.1), not included in the previous version