Aronszajn trees, square principles, and stationary reflection

Lambie-Hanson, Chris

doi:10.1002/malq.201600040

Cited by 13 publications

(19 citation statements)

References 23 publications

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“…Note that the choice of µ = 2 is justified by a recent result of Shani [Sha16] and independently Lambie-Hanson [LH17] which implies that none of the results obtained in this paper follow from µ = 3. )…”

Section: Introductionmentioning

confidence: 77%

Reduced Powers of Souslin Trees

Brodsky¹,

Rinot²

2017

Forum of Mathematics, Sigma

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We study the relationship between a κ-Souslin tree T and its reduced powers T θ / . Previous works addressed this problem from the viewpoint of a single power θ , whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an ℵ 6 -Souslin tree T and a sequence of uniform ultrafilters n | n < 6 such that T ℵn / n is ℵ 6 -Aronszajn if and only if n < 6 is not a prime number. This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

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Section: Introductionmentioning

confidence: 77%

Reduced Powers of Souslin Trees

Brodsky¹,

Rinot²

2017

Forum of Mathematics, Sigma

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“…In particular, we mention a recent result of Krueger [13] on the club-isomorphism of higher Aronszajn trees that could substitute the Abraham-Shelah model. The construction schemes developed by Brodsky, Lambie-Hanson and Rinot [4,15,18] for higher Suslin and Aronszajn trees also seem rather relevant. The theorem of Jensen on CH and all Aronszajn trees being special was recently generalized to higher cardinals in a breakthrough result by Asperó and Golshani [2].…”

Section: Proof (1) Suppose Thatmentioning

confidence: 99%

On the complexity of classes of uncountable structures: trees on $\aleph _1$

Friedman

Soukup

2021

Fund. Math.

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We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire space ω ω 1 1. First, we show that none of these classes have the Baire property (unless they are empty). Moreover, under V = L, (a) the class of Aronszajn and Suslin trees is Π 1 1-complete, (b) the class of special Aronszajn trees is Σ 1 1-complete, and (c) the class of Kurepa trees is Π 1 2-complete. We achieve these results by finding nicely definable reductions that map subsets X of ω1 to trees TX so that TX is in a given tree class T if and only if X is stationary/non-stationary (depending on the class T). Finally, we present models of CH where these classes have lower projective complexity. 1 and 2 ω 1. Basic open sets correspond to countable partial functions, which, in turn, give rise to an ω 1-Borel structure on ω ω 1 1 , 2 ω 1 and so P(ω 1) as well. This allows us to measure the complexity of subsets of ω ω 1 1 or, equivalently, of families of natural combinatorial structures on ω 1. In this paper, we will focus on models of CH, i.e., 2 ℵ 0 = ℵ 1. This is a fairly natural assumption in

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“…For example, if µ is singular, then µ implies ind µ,cf(µ) (cf. [7]), while, if λ < µ are regular infinite cardinals, then (µ) implies ind (µ, λ) (cf. [9]).…”

Section: Definition 34 ([8]mentioning

confidence: 99%

A note on highly connected and well-connected Ramsey theory

Lambie-Hanson¹

2020

Preprint

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We study a pair of weakenings of the classical partition relation ν → (µ) 2 λ recently introduced by Bergfalk-Hrušák-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on ν-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complementary positive and negative results indicating the effect of large cardinals, forcing axioms, and square principles on these partition relations. We also prove a consistency result indicating that a non-trivial instance of the stronger of these two partition relations can hold at the continuum.

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Aronszajn trees, square principles, and stationary reflection

Cited by 13 publications

References 23 publications

Reduced Powers of Souslin Trees

Reduced Powers of Souslin Trees

On the complexity of classes of uncountable structures: trees on $\aleph _1$

A note on highly connected and well-connected Ramsey theory

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