Comparing almost-disjoint families

Abstract

In this paper families of almost-disjoint, countably infinite sets will be considered, i.e. families of type ~r e~z) where [A.[=~o and tA~NAp] is finite whenever 0~</3<• One of the early results in the theory of almost-disjoint sets is the following theorem of E. W. Miller [3]: if d = (A~: e < x) is a family of infinite sets and instead of almost-disjointness the stronger ]A, NA~]<-n holds with a fixed n<co, then ~r has the property B, i.e. there is a coloring of the union set with two colors and without a monochromatic A. This is no longer true for almostdisjoint families.The role of property B in connection with this result is a bit misleading, because, as it was later discovered, much stronger properties hold under the Miller condition, e.g. d has a transversal -a one-to-one choice function. That the transversal property really implies property B, at least for families of infinite sets, is proved in [1]. The strongest property to date gained from the Miller condition is the following: d is essentially disjoint, i.e. there are finite sets (B~: ~<• such that the sets {A~-B~: a<x} are disjoint (see [2]). This property is obviously stronger than any of property B and the transversal property. As these properties are proved under a relatively mild restriction on the size of pairwise intersections, one may wonder whether one of them or any other nontrivial property at all can be recognised from the matrix {]A~NA~[: c~</?<~}. Call two enumerated almost-disjoint families d={A~: ~<~} and N={B,: c~<~} similar if ]A~NApl= [B~NBp] holds whenever c~<fl<z. The question therefore is : which properties are similarity-invariant ? It is easy to see that property B and the transversal property are not, as we can simultaneously enlarge each set with two (or infinitely many) new elements without changing any A~ NAp. However, essential-disjointness is similarity-invariant. Even more is true, namely a certain monotonicity holds: if d=(A~: ~<z) and ~=(B~: ~<z) are both almostdisjoint families, IA~NAp]<= ]B~NBp] holds for c~</~<z and B is essentially disjoint then so is d. The proof of this statement is the main content of the present note. We will derive from it, as an immediate corollary, Miller's theorem.In this paper the standard set theory notation is used.

Splitting families of sets in ZFC

“…More results with the GCH followed along this line [14,15,9,10]. Komjáth [14] proved from the GCH that for ν + < ρ every ρ-uniform F which satisfies C(ρ + , ν) and is also almost disjoint is essentially disjoint, that is, it can be made pairwise disjoint by removing a set of size < ρ from each member of F.…”

“…Komjáth [15] investigated further the property of essential disjointness which he introduced in [14] (under the name "sparseness") and proved that the array of cardinalities of pairwise intersections in an ℵ 0 -uniform a.d family determines whether the family is essentially disjoint or not.…”

“…Definition 4.12. Two families of sets F and G are similar if there is a bijection f :F → G such that |A ∩ B| = |f (A) ∩ f (B)| for all A, B ∈ F.Komjáth[15] addressed the following question: which combinatorial properties of ℵ 0 -uniform families are invariant under similarity? Property B and the existence of a disjoint refinement are not similarity invariant, as one can replace A ∈ F by A∪D(A) where D(A)∩D(B) = ∅ for distinct A, B ∈ F and obtain an equivalent family which has a disjoint refinement from a family which does not satisfy property B.…”

On the arithmetic of density

“…Recently, asymptotic results in infinite graph theory and in the combinatorics of families of sets [9,10] -some of which were proved earlier with the GCH or with forms of the SCH [5,3,6,11,12] -were proved in ZFC by making use of an eventual regularity property of density: that density satisfies a version of Shelah's RGCH theorem. See also [14] on the question whether the use of RGCH in [9] is necessary.…”