scholarly journals Drawings of Cm×Cn with One Disjoint Family II

2001 ◽  
Vol 82 (1) ◽  
pp. 161-165 ◽  
Author(s):  
Hector A. Juarez ◽  
Gelasio Salazar
Keyword(s):  
Disjoint Family

On the Maximum Number of Edges in a Triple System Not Containing a Disjoint Family of a Given Size

2012 ◽  
Vol 21 (1-2) ◽  
pp. 141-148 ◽  
Author(s):  
PETER FRANKL ◽  
VOJTECH RÖDL ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Triple System ◽  
Disjoint Family

In 1965 Erdős conjectured a formula for the maximum number of edges in a k-uniform n-vertex hypergraph without a matching of size s. We prove this conjecture for k = 3 and all s ≥ 1 and n ≥ 4s.

Transversals with Residue in Moderately Overlapping T(k)-Families of Translates

2009 ◽  
Vol 52 (3) ◽  
pp. 388-402
Author(s):  
Aladár Heppes
Keyword(s):  
Asymptotic Behavior ◽  
Compact Convex ◽  
Line Transversal ◽  
Type Result ◽  
Empty Interior ◽  
Disjoint Family ◽  
Straight Line ◽  
The Family ◽  
Centrally Symmetric ◽  
Line Meeting

AbstractLet K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. The integer k is the stabbing level of the family. Two translates Ki = K + ci and Kj = K + cj are said to be σ-disjoint if σK + ci and σK + cj are disjoint. A recent Helly-type result claims that for every σ > 0 there exists an integer k(σ) such that if a family of σ-disjoint unit diameter discs has property T(k)|k ≥ k(σ), then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval K. The asymptotic behavior of k(σ) for σ → 0 is considered as well.Katchalski and Lewis proved the existence of a constant r such that for every pairwise disjoint family of translates of an oval K with property T(3) a straight line can be found meeting all but at most r members of the family. In the second part of the paper σ-disjoint families of translates of K are considered and the relation of σ and the residue r is investigated. The asymptotic behavior of r(σ) for σ → 0 is also discussed.

Weakly normal closures of filters on Pκλ

1993 ◽  
Vol 58 (1) ◽  
pp. 55-63 ◽  
Author(s):  
Masahiro Shioya
Keyword(s):  
Regular Cardinal ◽  
Natural Generalization ◽  
Large Cardinal ◽  
The Other ◽  
Disjoint Family ◽  
Straightforward Generalization ◽  
Weak Normality ◽  
Natural Formulation

The study of filters on Pκλ started by Jech [5] as a natural generalization of that of filters on an uncountable regular cardinal κ. Several notions including weak normality have been generalized. However, there are two versions proposed as weak normality for filters on Pκλ. One is due to Abe [1] as a straightforward generalization of weak normality for filters on κ due to Kanamori [6] and the other is due to Mignone [8]. While Mignone's version is weaker than normality, Kanamori-Abe's version is not in general. In fact, Abe [2] has proved, generalizing Kanamori [6], that a filter is weakly normal in the sense of Abe iff it is weakly normal in the sense of Mignone and there exists no disjoint family of cfλ-many positive sets. Therefore Kanamori-Abe's version is essentially a large cardinal property and Mignone's version seems to be the most natural formulation of “weak” normality.In this paper, we study weak normality in the sense of Mignone. In [8], Mignone studies weak normality of canonically defined filters. We complement his chart and try to find the weakly normal closures of these filters (i.e., the minimal weakly normal filters extending them). Therefore our result is a natural refinement of Carr [4].It is now well known that combinatorics on Pκλ is not a naive generalization of that on κ. For example, Menas [7] showed that stationarity on Pκλ can be characterized by 2-dimensional regressive functions, but not by 1-dimensional ones when λ is strictly larger than κ. We show in terms of weak normality that combinatorics on Pκλ vary drastically with respect to cfλ.

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 .

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 Drawings of Cm×Cn with One Disjoint Family

1999 ◽  
Vol 76 (2) ◽  
pp. 129-135 ◽  
Author(s):  
Gelasio Salazar
Keyword(s):  
Disjoint 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 .

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