TREE FORCING AND DEFINABLE MAXIMAL INDEPENDENT SETS IN HYPERGRAPHS

Abstract We show that after forcing with a countable support iteration or finite product of Sacks splitting over L , every analytic hypergraph on Polish space admits $\mathbf {\Delta }^1_2$ maximal independent set. This extends an earlier result by Schrittesser (see [25]). As main application we get the consistency $\mathfrak {r} = \mathfrak {u} {i} \omega _2$ together existence $\Delta ^1_2$ ultrafilter, $\Pi ^1_1$ family, and Hamel basis. solves open problems Brendle, Fischer, Khomskii [5] author [23]. also in ZFC {d} \leq {i}_{cl}$ addressing another question from [5].