An order property for families of sets

Abstract

We develop the idea of a ^-ordering (where 6 is an infinite cardinal) for a family of infinite sets. A 0-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a ^-transversal, are simple consequences of A possessing a 0-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a 0-ordering. The restrictions involve either intersection conditions on A (the intersection of every A-size subfamily of A has size at most K) or a chain condition on A.