On the spectra of cardinalities of branches of Kurepa trees

Characterizing the spectra of cardinalities of branches of Kurepa trees

We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.

Essential Kurepa trees versus essential Jech---Kunen trees

By an omega_1 --tree we mean a tree of size omega_1 and height omega_1. An omega_1 --tree is called a Kurepa tree if all its levels are countable and it has more than omega_1 branches. An omega_1 --tree is called a Jech--Kunen tree if it has kappa branches for some kappa strictly between omega_1 and 2^{omega_1}. A Kurepa tree is called an essential Kurepa tree if it contains no Jech--Kunen subtrees. A Jech--Kunen tree is called an essential Jech--Kunen tree if it contains no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2^{omega_1}> omega_2 that there exist esse…

Descriptive properties of higher Kurepa trees

We use generalizations of concepts from descriptive set theory to study combinatorial objects of uncountable regular cardinality, focussing on higher Kurepa trees and the representation of the sets of cofinal branches through such trees as continuous images of function spaces. For different types of uncountable regular cardinals $κ$, our results provide a complete picture of all consistent scenarios for the representation of sets of cofinal branches through $κ$-Kurepa trees as retracts of the generalized Baire space ${}^κκ$ of $κ$. In addition, these results can be used to determine the consis…

Planting Kurepa trees and killing Jech-Kunen trees in a model by using one inaccessible cardinal

By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it has kappa many branches for some kappa strictly between omega_1 and 2^{omega_1}. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist Kurepa trees and there are no Jech--Kunen trees, (2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees with omega_3 many branches exist.

库雷帕树、连续像与完美集性质

Kurepa trees, continuous images, and perfect set properties

基于Lücke和Schlicht的工作，从更高描述集合论视角研究库雷帕树，证明与CH一致时存在ω₂-库雷帕树但其分支集非ω₂ω₂连续像，还构造模型并证明全树、超薄树相关一致性结果。

Kurepa trees and spectra of $\mathcal{L}_{ω_1,ω}$-sentences

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathc…