MAD SPECTRA

2015 ◽  
Vol 80 (3) ◽  
pp. 901-916
Author(s):  
SAHARON SHELAH ◽  
OTMAR SPINAS
Keyword(s):  
Complete Solution ◽  
Almost Disjoint ◽  
Image Position ◽  
Maximal Almost Disjoint ◽  
The Universe ◽  
Almost Disjoint Families

AbstractThe mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets ${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption $\vartheta ^{ < \vartheta } = \vartheta $, where $\vartheta = {\rm{min}}\left( {\rm {\cal A}} \right) $.

scholarly journals MADNESS IN VECTOR SPACES

2019 ◽  
Vol 84 (4) ◽  
pp. 1590-1611
Author(s):  
IIAN B. SMYTHE
Keyword(s):  
Ramsey Theory ◽  
Vector Spaces ◽  
Mad Families ◽  
Almost Disjoint ◽  
Infinite Cardinality ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

AbstractWe consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability.

scholarly journals Maximal almost disjoint families and pseudocompactness of hyperspaces

2021 ◽  
pp. 107872
Author(s):  
O. Guzmán ◽  
M. Hrušák ◽  
V.O. Rodrigues ◽  
S. Todorčević ◽  
A.H. Tomita
Keyword(s):  
Almost Disjoint ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

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.

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.

Continuum cardinals generalized to Boolean algebras

2001 ◽  
Vol 66 (4) ◽  
pp. 1928-1958 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Boolean Algebras ◽  
Infinite Cardinal ◽  
Cardinal Numbers ◽  
Almost Disjoint ◽  
Infinite Sets ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.

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