Partition Forcing and Independent Families

CRUZ-CHAPITAL, JORGE A.; Fischer, Vera; Guzmán, Osvaldo; Šupina, Jaroslav

doi:10.1017/jsl.2022.68

Cited by 3 publications

(10 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First we recall the poset Q I from [24] and some of its properties. The exposition in this section strongly mirrors that of [7,Section 5], where it is shown that Q I strongly preserves tight MAD families.…”

Section: §1 Introduction Two Infinite Subsets Of a B ∈ [ ] Are Almos...mentioning

confidence: 73%

“…(5) For each n ṫn is a P(K) name for a node in Ṫϕ 0 (n) extending the ϕ 1 (n)th-node in Ṫϕ 0 (n) . ( 6) For each n < kn is a name for an element of in M. (7) For each s ∈ n and i, m…”

Section: §1 Introduction Two Infinite Subsets Of a B ∈ [ ] Are Almos...mentioning

confidence: 99%

“…Remark 3. In [7] it was shown that in the iterated partition forcing model that a = i = u = ℵ 1 . Alongside Theorem 7.5 this result gives credence to the idea that the iterated partition forcing model is a model where every relative of a other than…”

Section: §1 Introduction Two Infinite Subsets Of a B ∈ [ ] Are Almos...mentioning

confidence: 99%

See 2 more Smart Citations

Tight Eventually Different Families

Fischer¹,

Switzer²

2023

J. symb. log.

Self Cite

View full text Add to dashboard Cite

Generalizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of $\omega $ . Such sets strengthen maximality, exist under $\mathsf {MA} (\sigma \mathrm {-centered})$ and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals $\mathfrak {a}_e$ and $\mathfrak {a}_p$ in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including $\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$ , $\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$ , and $\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$ . We also show that there are $\Pi ^1_1$ tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside $\Pi ^1_1$ witnesses for $\mathfrak {a}_e = \mathfrak {a}_p = \aleph _1$ . Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.

show abstract

Section: §1 Introduction Two Infinite Subsets Of a B ∈ [ ] Are Almos...mentioning

confidence: 73%

Section: §1 Introduction Two Infinite Subsets Of a B ∈ [ ] Are Almos...mentioning

confidence: 99%

See 1 more Smart Citation

Tight Eventually Different Families

Fischer¹,

Switzer²

2023

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A tree version of Shelah's poset, known as party forcing, has been used in [7] to establish the consistency of i = f < u, where f is the free sequences number. 3 For recent studies on Sacks indestructible, co-analytic maximal independent families see [5], as well as [1,9,23]. In this paper, we prove:…”

Section: §1 Introductionmentioning

confidence: 74%

“…Moreover we make an explicit use of an equivalent characterisation of dense maximality given in Lemma 6.2, characterization which plays a key role in our main theorem. An analogue to the countable setting of the overall approach, which we take in this paper can be found in the more recent studies [1,9,23]. Note that an analogue of the equivalent characterization given in Lemma 6.2 implicitly appears in [22].…”

Section: §1 Introductionmentioning

confidence: 94%

Higher Independence

Fischer¹,

MONTOYA²

2022

J. symb. log.

View full text Add to dashboard Cite

We study higher analogues of the classical independence number on $\omega $ . For $\kappa $ regular uncountable, we denote by $i(\kappa )$ the minimal size of a maximal $\kappa $ -independent family. We establish ZFC relations between $i(\kappa )$ and the standard higher analogues of some of the classical cardinal characteristics, e.g., $\mathfrak {r}(\kappa )\leq \mathfrak {i}(\kappa )$ and $\mathfrak {d}(\kappa )\leq \mathfrak {i}(\kappa )$ . For $\kappa $ measurable, assuming that $2^{\kappa }=\kappa ^{+}$ we construct a maximal $\kappa $ -independent family which remains maximal after the $\kappa $ -support product of $\lambda $ many copies of $\kappa $ -Sacks forcing. Thus, we show the consistency of $\kappa ^{+}=\mathfrak {d}(\kappa )=\mathfrak {i}(\kappa )<2^{\kappa }$ . We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence.

show abstract

Cohen preservation and independence

Fischer

Switzer

2023

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

Partition Forcing and Independent Families

Cited by 3 publications

References 52 publications

Tight Eventually Different Families

Tight Eventually Different Families

Higher Independence

Cohen preservation and independence

Contact Info

Product

Resources

About