Tree forcing and definable maximal independent sets in hypergraphs

Abstract

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a ${\mathbf{\Delta }}_2^1$ maximal independent set. As a main application we get the consistency of $\mathfrak{r}=\mathfrak{u}=\mathfrak{i}={\omega }_2$ together with the existence of a ${\Delta }_2^1$ ultrafilter, a ${\Pi }_1^1$ maximal independent family and a ${\Delta }_2^1$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d}\le {\mathfrak{i}}_{cl}$.