Strongs-Sequences and Variations on Martin's Axiom

1985 ◽  
Vol 37 (4) ◽  
pp. 730-746 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Family ◽  
Uncountable Family ◽  
Martin’S Axiom ◽  
Finite Set ◽  
Almost Disjoint ◽  
Disjoint Sets ◽  
Almost Disjoint Family ◽  
Require Equality ◽  
Single Set

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

scholarly journals An order property for families of sets

1988 ◽  
Vol 44 (3) ◽  
pp. 294-310
Author(s):  
N. H. Williams
Keyword(s):  
Pairwise Disjoint ◽  
Chain Condition ◽  
Infinite Cardinal ◽  
Order Property ◽  
Disjoint Family ◽  
Combinatorial Properties ◽  
The Family ◽  
Almost Disjoint ◽  
A Chain ◽  
Almost Disjoint Family

AbstractWe develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-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 θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrictions involve either intersection conditions on A (the intersection of every λ-size subfamily of A has size at most κ) or a chain condition on A.

Two step iteration of almost disjoint families

2004 ◽  
Vol 69 (1) ◽  
pp. 81-90 ◽  
Author(s):  
Jerry E. Vaughan
Keyword(s):  
Pairwise Disjoint ◽  
Infinite Family ◽  
Topological Spaces ◽  
Disjoint Family ◽  
Almost Disjoint ◽  
Countably Infinite ◽  
Almost Disjoint Family ◽  
Maximal Almost Disjoint ◽  
Infinite Set ◽  
Almost Disjoint Families

Let E be an infinite set, and [E]ω the set of all countably infinite subsets of E. A family ⊂ [E]ω is said to be almost disjoint (respectively, pairwise disjoint) provided for A, B ∈ , if A ≠ B then A ∩ B is finite (respectively, A ∩ B is empty). Moreover, an infinite family A is said to be a maximal almost disjoint family provided it is an infinite almost disjoint family not properly contained in any almost disjoint family. In this paper we are concerned with the following set of topological spaces defined from (maximal) almost disjoint families of infinite subsets of the natural numbers ω.

scholarly journals Families of partial representing sets

1985 ◽  
Vol 38 (2) ◽  
pp. 198-206
Author(s):  
Kevin P. Balanda
Keyword(s):  
Disjoint Family ◽  
Empty Intersection ◽  
The Family ◽  
Almost Disjoint ◽  
Almost Disjoint Family

AbstractAssume GCH. Let κ, μ, Σ be cardinals, with κ infinite. Let be a family consisting of λ pairwise almost disjoint subsets of Σ each of size κ, whose union is Σ. In this note it is shown that for each μ with 1 ≤ μ ≤min(λ, Σ), there is a “large” almost disjoint family of μ-sized subsets of Σ, each member of having non-empty intersection with at least μ members of the family .

scholarly journals A special class of almost disjoint families

1995 ◽  
Vol 60 (3) ◽  
pp. 879-891 ◽  
Author(s):  
Thomas E. Leathrum
Keyword(s):  
Ordered Sets ◽  
Minimum Cardinality ◽  
Disjoint Family ◽  
Open Questions ◽  
Consistency Results ◽  
The Family ◽  
Almost Disjoint ◽  
Cohen Model ◽  
Almost Disjoint Family ◽  
Almost Disjoint Families

AbstractThe collection of branches (maximal linearly ordered sets of nodes) of the tree <ωω (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal — for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and is the minimum cardinality of a maximal off-branch family.Results concerning include: (in ZFC) , and (consistent with ZFC) is not equal to any of the standard small cardinal invariants or = 2ω. Most of these consistency results use standard forcing notions—for example, in the Cohen model.Many interesting open questions remain, though—for example, whether .

Families of sets whose pairwise intersections have prescribed cardinals or order types Corrigenda

1977 ◽  
Vol 81 (3) ◽  
pp. 523-523
Author(s):  
P. Erdös ◽  
E. C. Milner ◽  
R. Rado
Keyword(s):  
Disjoint Family ◽  
Counter Example ◽  
Order Types ◽  
Almost Disjoint ◽  
Image Position ◽  
Almost Disjoint Family

(i) J. Baumgartner has kindly drawn our attention to the fact that Theorem 2 as stated in (1) is false. A counter example is the case in which m = ℵ2; n = ℵ1; p = ℵ0. For by reference (3) of the paper (1) there is an almost disjoint family (Aγ: γ < ω1) of infinite subsets of ω̲ Put Aν = ω̲ for ω1 ≤ ν < ω2. Then, contrary to the assertion of that theorem, all conditions of Theorem 2 are satisfied. However, Theorem 2 becomes correct if the hypothesisis strengthened toIn fact, Baumgartner has proved the desired conclusion under the weaker hypothesis

On representing sets of an almost disjoint family of sets

1987 ◽  
Vol 101 (3) ◽  
pp. 385-393
Author(s):  
P. Komjath ◽  
E. C. Milner
Keyword(s):  
Cardinal Number ◽  
Disjoint Family ◽  
Cardinal Numbers ◽  
Almost Disjoint ◽  
Image Position ◽  
Almost Disjoint Family

For cardinal numbers λ, K, ∑ a (λ, K)-family is a family of sets such that || = and |A| = K for every A ε , and a (λ, K, ∑)-family is a (λ,K)-family such that |∪| = ∑. Two sets A, B are said to be almost disjoint ifand an almost disjoint family of sets is a family whose members are pairwise almost disjoint. A representing set of a family is a set X ⊆ ∪ such that X ∩ A = ⊘ for each A ε . If is a family of sets and |∪| = ∑, then we write εADR() to signify that is an almost disjoint family of ∑-sized representing sets of . Also, we define a cardinal number

Maximally almost disjoint families of representing sets

1983 ◽  
Vol 93 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Kevin P. Balanda
Keyword(s):  
Disjoint Family ◽  
Empty Intersection ◽  
The Family ◽  
Almost Disjoint ◽  
Almost Disjoint Family ◽  
Almost Disjoint Families

A family of κ-sized sets is said to be almost disjoint if each pair of sets from the family intersect in a set of power less than κ. Such an almost disjoint family ℋ is defined to be κ-maximally almost disjoint (κ-MAD) if |∪ℋ| = κ and each κ-sized subset of ∪ ℋ intersects some member of ℋ in a set of cardinality κ. A set T is called a representing set of a family if T ⊆ ∪ and T has non-empty intersection with each member of .

Almost disjoint sets and Martin's axiom

1979 ◽  
Vol 44 (3) ◽  
pp. 313-318 ◽  
Author(s):  
Michael L. Wage
Keyword(s):  
Partial Order ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Almost Disjoint ◽  
Disjoint Sets

AbstractWe present a number of results involving almost disjoint sets and Martin's axiom. Included is an example, due to K. Kunen, of a c.c.c. partial order without property K whose product with every c.c.c. partial order is c.c.c.

Adjoining cofinitary permutations

1999 ◽  
Vol 64 (4) ◽  
pp. 1803-1810 ◽  
Author(s):  
Yi Zhang
Keyword(s):  
Proper Subset ◽  
Disjoint Family ◽  
Almost Disjoint ◽  
Almost Disjoint Family ◽  
Maximal Almost Disjoint

AbstractWe show that it is consistent with ZFC + ¬CH that there is a maximal cofinitary group (or, maximal almost disjoint group) G ≤ Sym(ω) such that G is a proper subset of an almost disjoint family A ⊆ Sym(ω) and ‖G‖ < ‖A‖. We also ask several questions in this area.

On the existence of strong chains in ℘(ω1)/Fin

1998 ◽  
Vol 63 (3) ◽  
pp. 1055-1062 ◽  
Author(s):  
Piotr Koszmider
Keyword(s):  
The Other ◽  
Disjoint Family ◽  
Other Hand ◽  
Chang’S Conjecture ◽  
Almost Disjoint ◽  
Chang's Conjecture ◽  
Almost Disjoint Family

Abstract(Xα: α < ω2) ⊂ ℘(ω1) is a strong chain in ℘(ω1)/Fin if and only if Xβ – Xα is finite and Xα – Xβ is uncountable for each β < α < ω1. We show that it is consistent that a strong chain in ℘(ω1) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in ℘(ω1) but no strong chain exists: is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain.

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