scholarly journals The Boolean prime ideal theorem does not imply the extension of almost disjoint families to MAD families

2021 ◽  
Author(s):  
Eleftherios Tachtsis
Keyword(s):  
Prime Ideal ◽  
Mad Families ◽  
Almost Disjoint ◽  
Almost Disjoint Families ◽  
Prime Ideal Theorem

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.

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.

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

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.

Comparing almost-disjoint families

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