ALMOST DISJOINT FAMILIES OF COUNTABLE SETS

Abstract

This chapter elaborates the almost disjoint families of countable sets. A family of countable subsets of a given infinite set X is almost disjoint, if any two distinct members meet in a finite set only. A family F ⊆ P(X) has an almost disjoint refinement provided that there is an almost disjoint family A of countable sets such that each member of F contains some member of A. Objects of study are the families having an almost disjoint refinement (ADR). The most exciting problem in this area may is stated. Given a maximal almost disjoint family A on N, the set of all natural numbers (also denoted by ω), the existence of an almost disjoint refinement for the family of all large sets with respect to A, those that meet in an infinite set infinitely many members of A is analyzed. The approach is based on cardinal characteristics of sets and functions, which enables us to prove some special cases of the problem. The results are either proved in ZFC only or formulated with help of these cardinal characteristics.