Splitting Strongly Almost Disjoint Families

Abstract

ABSTRACT. We say that a family A C [A]K is strongly almost disjoint if something more than just \A n B\ < n, e.g. that \A f! B\ < a < k, is assumed for A, B G A. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each A E A there is F(A) G [A]<K so that {A \ F(A) : A G A} is disjoint.On the other hand, we get from a supercompact cardinal the consistency of GCH plus the existence of a family A C [ww-|-i]wi whose elements have pairwise finite intersections and such that it does not even have property B. This solves an old problem raised in [4]. The same example is also used to produce a graph of chromatic number 012 on u)u+i that does not contain [w,w], answering a problem from [5].We also have applications of our results to "splitting" certain families of closed subsets of a topological space. These improve results from [3, 12 and 13]. Introduction.One of the oldest results of set theory applied to topology is Bernstein's theorem [2] which says that the real line R can be split into two disjoint parts, say Xo and X\, in such a way that every Cantor set in R (or, equivalently, every uncountable closed subset of R) intersects both Xo and Xi. Using the terminology introduced in [8], this means that the family C(R) of all Cantor sets in R has property B.There is another, by now classical, result due to Miller [8] which states that if A is any strongly almost disjoint family of countably infinite sets (here "strongly almost disjoint" means that there is a n G oj such that if A, B G A and A^ B then \A fi B\ < n), then A has property B.In this paper we propose to show that both of these classical results, as well as numerous further strengthenings and generalizations of them, boil down to the same thing. More precisely, we will show that all of these results (see e.g. [3,4,7,8,12,13]) are relatively easy consequences of a single general theorem that we call the Main Theorem. It should not come then as a surprise that such a general result has a somewhat complicated formulation.The two classical results we mentioned above are, of course, proved in ZFC, but some of the strengthenings and/or generalizations did require assumptions going beyond ZFC. We shall also present here several results that show that some of these extra set-theoretic hypotheses are indeed essential. Thus we shall solve problems raised to this effect, e.g., in [4].1. The Main Theorem.Our set-theoretic notation is standard. We shall list below some (really simple) definitions, notations and terminology that we need to give a transparent formulation of our Main Theorem.

Namba Forcing and No Good Scale

“…Magidor [1] showed that the same conclusion follows from Martin's Maximum. Finally, Cummings, Foreman, and Magidor [5] proved that a model of Shelah [9] in which the approachability property AP ℵω fails also satisfies that there is no good scale.…”

Transversals for strongly almost disjoint families

“…The paper [3] is a natural sequel to the paper of Erdös and Hajnal. It answers a number of questions from [2] and considers conditions when almost disjoint families A ⊆ [λ] κ are essentially disjoint: removing a set of size κ from each member produces a disjoint family.…”

Strongly almost disjoint sets and weakly uniform bases

“…We distinguish two cases. 3 Note that in this argument, as well as in the proof of the next lemma, only the following consequence of weak closedness is used:…”

“…It turns out that CECA is equivalent to GCH + some previously known weakenings of 2 λ , but CECA has a different flavor than 2 λ -principles and may be easier to work with. The equivalence will be shown in Section 1.In [3] it is shown that under certain conditions almost disjoint families {A α : α < κ} can be refined to disjoint families by removing small sets A α from each A α . Let us say that a family {A α : α < κ} is point-< τ if for every I ∈ [κ] τ the intersection α∈I A α is empty.…”

“…In [3] it is shown that under certain conditions almost disjoint families {A α : α < κ} can be refined to disjoint families by removing small sets A α from each A α . Let us say that a family {A α : α < κ} is point-< τ if for every I ∈ [κ] τ the intersection α∈I A α is empty.…”