How to drive our families mad

TL;DR

This paper investigates the minimal size of maximal almost disjoint families extending a given family on a countable set, exploring their possible cardinalities and consistency results related to the continuum.

Contribution

It introduces the concept of $a^+(F)$ for almost disjoint families and demonstrates the range of cardinalities these can attain, including consistency results involving the continuum.

Findings

All infinite cardinals up to the continuum can be realized as $a^+(F)$ for some almost disjoint family.

The inequalities $eth_1=a<a^+(eth_1)=c$ and $a=a^+(eth_1)<c$ are consistent.

Constructed mad families with additional specified properties.

Abstract

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(\kappa)=\sup\{a^+(F): |F| \leq \kappa \}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional properties.