Maximal discrete sets

TL;DR

This paper surveys the definability and size properties of maximal discrete sets in analytic hypergraphs, covering various examples like mad families and cofinitary groups, and discusses related forcing techniques.

Contribution

It provides a comprehensive overview of the current results on maximal discrete sets, including new insights into spectra of characteristics and forcing methods.

Findings

Analysis of the size and definability of maximal almost disjoint families

Discussion of spectra of cardinal characteristics

Description of Zhang's forcing for adding generic cofinitary permutations

Abstract

We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_\xi$, for $\xi<\omega_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.