Cohen-stable families of subsets of integers

2001 ◽  
Vol 66 (1) ◽  
pp. 257-270 ◽  
Author(s):  
Miloš S. Kurilić
Keyword(s):  
Countable Family ◽  
Generic Extension ◽  
Mad Families ◽  
Stable Splitting ◽  
Almost Disjoint ◽  
Maximal Almost Disjoint ◽  
Infinite Set ◽  
Families Of Subsets

AbstractA maximal almost disjoint (mad) family ⊆ [ω]ω is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family. .is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A]. A ∈ are nowhere dense. An ℵ0-mad family, . is a mad family with the property that given any countable family ℬ ⊂ [ω]ω such that each element of ℬ meets infinitely many elements of in an infinite set there is an element of meeting each element of ℬ in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ0-mad families. Either of the conditions b = c or a < cov() implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family. . is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈ are nowhere dense. Also. Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ0-splitting families of cardinality ≤ κ exist.

scholarly journals The Ramsey property implies no mad families

2019 ◽  
Vol 116 (38) ◽  
pp. 18883-18887 ◽  
Author(s):  
David Schrittesser ◽  
Asger Törnquist
Keyword(s):  
Set Theory ◽  
Large Set ◽  
Descriptive Set Theory ◽  
Ramsey Property ◽  
Mad Families ◽  
Almost Disjoint ◽  
Descriptive Set ◽  
Maximal Almost Disjoint ◽  
Borel Ideals ◽  
Standing Problem

We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

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.

GENERIC EXISTENCE OF MAD FAMILIES

2017 ◽  
Vol 82 (1) ◽  
pp. 303-316 ◽  
Author(s):  
OSVALDO GUZMÁN-GONZÁLEZ ◽  
MICHAEL HRUŠÁK ◽  
CARLOS AZAREL MARTÍNEZ-RANERO ◽  
ULISES ARIET RAMOS-GARCÍA
Keyword(s):  
Combinatorial Properties ◽  
Mad Families ◽  
Almost Disjoint ◽  
Generic Existence ◽  
Maximal Almost Disjoint

AbstractIn this note we study generic existence of maximal almost disjoint (MAD) families. Among other results we prove that Cohen-indestructible families exist generically if and only if b = c. We obtain analogous results for other combinatorial properties of MAD families, including Sacks-indestructibility and being +-Ramsey.

Ordering MAD families a la Katětov

2003 ◽  
Vol 68 (4) ◽  
pp. 1337-1353 ◽  
Author(s):  
Michael Hrušák ◽  
Salvador García Ferreira
Keyword(s):  
Continuum Hypothesis ◽  
Mad Families ◽  
Almost Disjoint ◽  
Maximal Almost Disjoint ◽  
The Continuum

AbstractAn ordering (≤K) on maximal almost disjoint (MAD) families closely related to destructibility of MAD families by forcing is introduced and studied. It is shown that the order has antichains of size c and decreasing chains of length c+ bellow every element. Assuming t = c a MAD family equivalent to all of its restrictions is constructed. It is also shown here that the Continuum Hypothesis implies that for every ωω-bounding forcing ℙ of size c there is a Cohen-destructible, ℙ-indestructible MAD family. Finally, two other orderings on MAD families are suggested and an old construction of Mrówka is revisited.

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 Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

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 Mad families, P+(I)-ideals and ideal convergence

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3099-3108
Author(s):  
Jiakui Yu ◽  
Shuguo Zhang
Keyword(s):  
Necessary Condition ◽  
Ideal Convergence ◽  
Mad Families ◽  
Helly Property ◽  
Almost Disjoint ◽  
Almost Disjoint Families ◽  
The Ideal

Let I be an ideal on ?, the notion of I-AD family was introduced in [3]. Analogous to the well studied ideal I(A) generated by almost disjoint families, we introduce and investigate the ideal I(I-A). It turns out that some properties of I(I-A) depends on the structure of I. Denoting by a(I) the minimum of the cardinalities of infinite I-MAD families, several characterizations for a(I) ? ?1 will be presented. Motivated by the work in [23], we introduce the cardinality s?,?(I), and obtain a necessary condition for s?,?(I) = s(I). As an application, we show finally that if a(I) ? s(I), then BW property coincides with Helly property.

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