Tree forcing and definable maximal independent sets in hypergraphs

TL;DR

This paper demonstrates that certain forcing methods ensure the existence of definable maximal independent sets in hypergraphs on Polish spaces, leading to new consistency results involving cardinal characteristics and definable sets.

Contribution

It establishes the existence of definable maximal independent sets in hypergraphs after specific forcing, solving open problems related to definability and cardinal invariants.

Findings

Existence of $oldsymbol{ riangle}^1_2$ maximal independent sets after forcing.

Consistency of $rak{r}=rak{u}=rak{i}=oldsymbol{\omega_2}$ with definable ultrafilters and bases.

ZFC result that $rak{d} leq rak{i}_{cl}$.

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 $\mathbf{\Delta}^1_2$ 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 $\Delta^1_2$ ultrafilter, a $\Pi^1_1$ maximal independent family and a $\Delta^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.