Mad Families and Their Neighbors

We study several sorts of maximal almost disjoint families, both on a countable set and on uncountable, regular cardinals. We relate the associated cardinal invariants with bounding and dominating numbers and also with the uniformity of the meager ideal and some of its generalizations. 1. Who Are These Families? A Background Check Almost disjoint (ad) families have been a topic of interest in set theory since its early days [20, 23]. Once the forcing technique became available, a great deal was learned about their properties, in particular the properties of maximal almost disjoint (mad) families of sets of natural numbers; see for example [14, 16]. A recent major breakthrough is Shelah’s proof [19] (see also [5]) that the minimum size of a mad family may be larger than the dominating number. In this paper, we investigate some properties of mad families, not only of sets but also of functions and of permutations. Although part of our work (Section 5) is concerned with mad families on the set ω of natural numbers, most of what we do is in the context of arbitrary regular cardinals. Indeed, some of our results extend to the uncountable case results already known for ω, while others exhibit differences in the uncountable case from the known facts for ω. We begin by defining the notation we shall use and recalling some information about mad families and certain other families of sets and functions (the neighbors mentioned in the title). We use standard set-theoretic notation and terminology as in [13, 14]. Convention 1.1. Throughout the paper, κ is an infinite, regular cardinal. Definition 1.2. Two sets are κ-almost disjoint (κ-ad) if their intersection has cardinality strictly smaller than κ. A family of sets is κ-ad if every two distinct members of it are κ-ad. Definition 1.3. A maximal almost disjoint (mad) family of subsets of κ is a κ-ad family of unbounded subsets of κ that has size at least κ and that is not properly included in another such family. a(κ) is the smallest cardinality of any such family. Observe that the maximality clause in the definition means that every unbounded subset of κ intersects some member of the mad family in an unbounded set. The requirement that a mad family have size at least κ is needed to exclude trivial examples, like a partition of κ into fewer than κ unbounded pieces. It implies that 2000 Mathematics Subject Classification. 03E05. Blass was partially supported by the United States National Science Foundation, grant DMS–