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 ∆ 1 2 maximal independent set. This extends an earlier result by Schrittesser (see [19]). As a main application we get the consistency of r = u = i = ω 2 together with the existence of a ∆ 1 2 ultrafilter, a Π 1 1 maximal independent family and a ∆ 1 2 Hamel basis. This solves open problems of Brendle, Fischer and Khomskii [3] and the author [18]. We also show in ZFC that d ≤ i cl , adressing another question from [3].