Invariance Properties of Almost Disjoint Families

2013 ◽  
Vol 78 (3) ◽  
pp. 989-999 ◽  
Author(s):  
M. Arciga-Alejandre ◽  
M. Hrušák ◽  
C. Martinez-Ranero
Keyword(s):  
Invariance Properties ◽  
Katětov Order ◽  
Almost Disjoint ◽  
Almost Disjoint Families

AbstractWe answer a question of Garcia-Ferreira and Hrušák by consistently constructing a MAD family maximal in the Katětov order. We also answer several questions of Garcia-Ferreira.

scholarly journals On Weakly Tight Families

2012 ◽  
Vol 64 (6) ◽  
pp. 1378-1394 ◽  
Author(s):  
Dilip Raghavan ◽  
Juris Steprāns
Keyword(s):  
Disjoint Family ◽  
Katětov Order ◽  
Almost Disjoint ◽  
Almost Disjoint Family ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

Abstract Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when 𝔠 < ℵω, we construct a weakly tight family under the hypothesis 𝔰 ≤ 𝔟 < ℵω. The case when 𝔰 < 𝔟 is handled in ZFC and does not require 𝔟 < ℵω, while an additional PCF type hypothesis, which holds when 𝔟 < ℵω is used to treat the case 𝔰 = 𝔟. The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

ALMOST DISJOINT FAMILIES OF COUNTABLE SETS

1984 ◽  
pp. 59-88 ◽  
Author(s):  
B. BALCAR ◽  
J. DOČKÁLKOVÁ ◽  
P. SIMON
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families

Comparing almost-disjoint families

1986 ◽  
Vol 47 (3-4) ◽  
pp. 321-323 ◽  
Author(s):  
P. Komjáth
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families

Fréchet-like properties and almost disjoint families

2020 ◽  
Vol 277 ◽  
pp. 107216
Author(s):  
César Corral ◽  
Michael Hrušák
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families

Almost Disjoint Families and Diagonalizations of Length Continuum

2010 ◽  
Vol 16 (2) ◽  
pp. 240-260
Author(s):  
Dilip Raghavan
Keyword(s):  
Cardinal Invariants ◽  
Almost Disjoint ◽  
Almost Disjoint Families ◽  
The Continuum

AbstractWe present a survey of some results and problems concerning constructions which require a diagonalization of length continuum to be carried out, particularly constructions of almost disjoint families of various sorts. We emphasize the role of cardinal invariants of the continuum and their combinatorial characterizations in such constructions.

Mad Families Constructed from Perfect Almost Disjoint Families

2013 ◽  
Vol 78 (4) ◽  
pp. 1164-1180 ◽  
Author(s):  
Jörg Brendle ◽  
Yurii Khomskii
Keyword(s):  
Cardinal Invariant ◽  
Mad Families ◽  
Almost Disjoint ◽  
Almost Disjoint Families

AbstractWe prove the consistency of together with the existence of a -definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an ℵ1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of (and hence, ).

scholarly journals Almost disjoint families and ultrapowers

2021 ◽  
Vol 13 ◽  
Author(s):  
Michalis Anoussis ◽  
Vaggelis Felouzis ◽  
Konstantinos Tsaprounis
Keyword(s):  
Banach Spaces ◽  
Alternative Characterization ◽  
Almost Disjoint ◽  
Measurable Cardinals ◽  
Almost Disjoint Families

We prove estimates for the cardinality of set-theoretic ultrapowers in terms of the cardinality of almost disjoint families. Such results are then applied to obtain estimates for the density of ultrapowers of Banach spaces. We focus on the change of the behavior of the corresponding ultrapower when certain ‘‘completeness thresholds’’ of the relevant ultrafilter are crossed. Finally, we also provide an alternative characterization of measurable cardinals.

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