Splitting, Bounding, and Almost
Disjointness Can Be Quite Different

Fischer, Vera; Mejía, Diego

doi:10.4153/cjm-2016-021-8

Cited by 12 publications

(7 citation statements)

References 18 publications

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“…For instance, It seems natural to expect that similar 3D-systems of iterations can be helpful in providing models in which, for example, b, s, and a are pairwise distinct. There are three ZFC admissible constellations: s < b < a, b < s < a, and b < a < s. The consistency of s = ℵ 1 < b < a holds in Shelah's original template model [23], while the consistency of ℵ 1 < s < b < a has been obtained by the first and third author of the current paper using the iteration of nondefinable (i.e., not Suslin) posets along a Shelah template (see [12]). The consistency of b < s < a (assuming the existence of a supercompact cardinal) is due to D. Raghavan and S. Shelah [22], and has been recently announced at the Oberwolfach Set Theory Meeting, February 2017.…”

Section: Definition 23 (Judah and Shelahmentioning

confidence: 77%

Coherent Systems of Finite Support Iterations

Fischer¹,

Friedman²,

Mejía³

et al. 2018

J. symb. log.

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We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ∆ 1 3 well-order of the reals. IntroductionIn this paper, we provide a generalization of the method of matrix iteration, to which we refer as 3D-coherent systems of iterations and which can be considered a natural extension of the matrix method to include a third dimension. That is, if a matrix iteration can be considered as a system of partial orders P α,β : α ≤ γ, β ≤ δ such that whenever α ≤ α and β ≤ β then P α,β is a complete suborder of P α ,β , then our 3D-coherent systems are systems of posets P α,β,ξ :As an application of this method, we construct models where Cichoń's diagram is separated into different values, one of them with 7 different values. Moreover, these models determine the value of a, which is actually the same as the value of b, and we further show that such models can be produced so that they satisfy, additionally, the existence of a ∆ 1 3 well-order of the reals. The method of matrix iterations, or 2D-coherent systems of iterations in our terminology, has already a long history. It was introduced by Blass and Shelah in [BS89], to show that consistently u < d, where u is the ultrafilter number and d is the dominating number. The method was further developed in [BF11], where the terminology matrix iteration appeared for the first time, to show that if κ < λ are arbitrary regular uncountable cardinals then there is a generic extension in which a = b = κ < s = λ. Here a, b and s denote the almost disjointness, bounding and splitting numbers respectively. In [BF11], the authors also introduce a new method for the preservation of a mad (maximal almost disjoint) family along a matrix iteration, specifically a mad family added by H κ (Hechler's poset for adding a mad family, see Definition 4.1), a method which is of particular importance for our current work. Later, classical preservation properties for matrix iterations were improved by Mejía [Mej13a] to provide several examples of models where the cardinals in Cichoń's diagram assume many different values, in particular, a model with 6 different values. Since then, the question of how many distinct values there can be simultaneously in Cichoń's diagram has been of interest for many authors, see for example [FGKS] (a 2010 Mathematics Subject Classification. 03E17, 03E15, 03E35, 03E40, 03E45.

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Section: Definition 23 (Judah and Shelahmentioning

confidence: 77%

Coherent Systems of Finite Support Iterations

Fischer¹,

Friedman²,

Mejía³

et al. 2018

J. symb. log.

Self Cite

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show abstract

“…The consistency of ω 1 < d < a and ω 1 < u < a was obtained by Shelah in [58] where he developed the technique of forcing along a template (see also [8] and [10]). In [24] Fischer and Mejía proved that it is consistent that ω 1 < s < b < a (see also [45] and [22]). Note that a positive solution to the question of Brendle and Raghavan would provide a positive solution to the problem of Roitman.…”

Section: Introductionmentioning

confidence: 83%

The ultrafilter and almost disjointness numbers

Guzmán¹,

Kalajdzievski²

2020

Preprint

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“…It is also known that s is independent from each of b and a individually. Some constellations of all three are also known to be consistent (see, e.g., [7]). The constructions tend to be difficult.…”

Section: Let Us Writementioning

confidence: 99%

Maximal discrete sets

Schrittesser¹

2020

Preprint

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We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, I-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics a ξ , for ξ < ω 1 as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.

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Splitting, Bounding, and Almost Disjointness Can Be Quite Different

Abstract: Abstract. We prove the consistency ofwith ZFC, where each of these cardinal invariants assume arbitrary uncountable regular values.

Cited by 12 publications

References 18 publications

Coherent Systems of Finite Support Iterations

Coherent Systems of Finite Support Iterations

The ultrafilter and almost disjointness numbers

Maximal discrete sets

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