There is a Van Douwen MAD family

TL;DR

This paper proves the existence of a maximal almost disjoint family of functions in omega^omega within ZFC, answering a long-standing question, and explores the limitations of analytic MAD families.

Contribution

It establishes the existence of a Van Douwen MAD family in ZFC and shows that analytic MAD families of functions are highly constrained or nonexistent.

Findings

Existence of a MAD family of functions in omega^omega in ZFC.

Strong constraints on the structure of analytic MAD families.

Nonexistence of analytic strongly MAD families of functions.

Abstract

We prove in ZFC that there is a MAD family of functions in omega^omega which is also maximal with respect to infinite partial functions. This solves a 20 year old question of Van Douwen. We also strengthen a result of J. Steprans stating that strongly MAD families of functions cannot be analytic. We show that analytic MAD families of functions, if they exist, must satisfy some strong constraints.