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The generalized Kurepa hypothesis at singular cardinals [article]

Mohammad Golshani
2020 arXiv   pre-print
We discuss the generalized Kurepa hypothesis KH_λ at singular cardinals λ.  ...  In particular, we answer questions of Erdös-Hajnal [1] and Todorcevic [6], [7] by showing that GCH does not imply KH_ℵ_ω nor the existence of a family F⊆ [ℵ_ω]^ℵ_0 of size ℵ_ω+1 such that F X has size  ...  introduction For an infinite cardinal λ let the generalized Kurepa hypothesis at λ, denoted KH λ , be the assertion: there exists a family F ⊆ P (λ) such that |F | > λ but |F ↾ X| ≤ |X| for every infinite  ... 
arXiv:1712.02487v2 fatcat:fruhjmqvi5da7aevxmg5iro2wy

A theorem on the closure of Ω-pure subgroups of CΩ groups in the Ω-topology

Patrick W Keef
1989 Journal of Algebra  
If we assume the continuum hypothesis (2%= H,), we can exhibit a specific example of a group which is close to being a dsc, but fails to be a dsc as follows: By [RW] there. is a balanced subgroup of a  ...  As was pointed out in the proof of Theorem 15, if we define Tar(G) = G. Theorems 15 and 17 are valid when n = 1.  ...  The main result of [K3] is that the existence of K-Kurepa subgroups is equivalent to the existence of families of sets of cardinality K satisfying Kurepa's Hypothesis (KH) .  ... 
doi:10.1016/0021-8693(89)90298-6 fatcat:3sj2t3iecraunbvzq37fx23yme

Page 436 of Mathematical Reviews Vol. 34, Issue 3 [page]

1967 Mathematical Reviews  
Throughout most of our work we assume the general continuum hypothesis. “In § 17 we prove a negative result on II-relations which is sharper than that obtained by Erdés and Hajnal [Acta Math. Acad.  ...  Independently of Erdés and Rado and at about the same time, Kurepa found several I- relations, the most important of which, deduced under the assumption of the generalized continuum hypothesis, can be  ... 

Page 5795 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
Hugh (1-CA) The generalized continuum hypothesis can fail everywhere. Ann. of Math. (2) 133 (1991), no. 1, 1-35.  ...  The subject matter need not be a unique, Platonic realm. This kind of generality is taken as one of the two essential and defining principles.  ... 

Book Review: Aspects of constructibility

Menachem Magidor
1975 Bulletin of the American Mathematical Society  
Besides, L satisfies the generalized continuum hypothesis, i.e. 2 Xot =N a +i for every infinite cardinal K«.  ...  (That includes Souslin Problem, Kurepa Hypothesis, different partition problems, two cardinals problems in model theory, etc.)  ...  The proof of Kurepa Hypothesis in L again extracts the right combinatorial property (0 + in this case).  ... 
doi:10.1090/s0002-9904-1975-13860-2 fatcat:avigu72rjndfflrxfshjdcu7qe

On Kurepa's problems in number Theory [article]

Aleksandar Ivić, Žarko Mijajlović
2003 arXiv   pre-print
We discuss some problems in number theory posed by Djuro Kurepa (1907-1993), including his classical left factorial hypothesis that an odd prime p does not divide 0! + 1! + ... + (p-1)!.  ...  This conjecture, known as the left factorial hypothesis, is still an open problem in number theory. There are several results which support the truth of the hypothesis.  ...  Further results on K(z) as a function of the complex variable z were obtained by Slavić [Sl] . His main result is that K(z) = − π e cotgπz + 1 e ∞ k=1 1 n!  ... 
arXiv:math/0312202v1 fatcat:xzxejlagerb23hm72lbc6jizze

On the History of Souslin's Problem

Carlos Alvarez
1999 Archive for History of Exact Sciences  
Kurepa 1935, p. 109. 85 "According to the results given in the last paragraphs, the hypothesis that every ramified table of power ℵ 1 is normal (or, equivalently, that every distinguished sequence of  ...  there is an F-generic filter on P .  ... 
doi:10.1007/s004070050038 fatcat:s4uvjfyvxvgdbgv6i55wi6m7fy

Automorphisms of $\omega \sb{1}$-trees

Thomas J. Jech
1972 Transactions of the American Mathematical Society  
.); by the induction hypothesis, T |aDT|a. We consider three cases. Case I. For some ß, a. = ß + 1.  ...  Here we generalize this result to show, by the way of consistency, that o(T) can be anything it is allowed to be. First we prove a refinement of Theorem 1 to show what o(T) is allowed to be.  ...  without proving the existence of a rigid tree. We conclude with the following problem; we are inclined rather to believe in the negative solution: Problem.  ... 
doi:10.1090/s0002-9947-1972-0347605-1 fatcat:dwiazgmy4ve6lckq73lnyxswwe

Partitioning pairs of countable sets

Stevo Todorčević
1991 Proceedings of the American Mathematical Society  
We make the translations of our partitions from [3] in the context of all countable subsets of a fixed uncountable set. A different translation was obtained recently by Velleman [4] .  ...  The proof will use straightforward generalization of one of the partitions from [3] .  ...  Since the family of singletons of elements of A is clearly a Kurepa family in [A] °, the hypothesis of the second theorem is satisfied when the stationary cofinality of [A]*0 is equal to the size of A  ... 
doi:10.1090/s0002-9939-1991-1036992-4 fatcat:35lhydg26vea7jl5jyfkdhvl74

Partitioning Pairs of Countable Sets

Stevo Todorcevic
1991 Proceedings of the American Mathematical Society  
We make the translations of our partitions from [3] in the context of all countable subsets of a fixed uncountable set. A different translation was obtained recently by Velleman [4] .  ...  The proof will use straightforward generalization of one of the partitions from [3] .  ...  Since the family of singletons of elements of A is clearly a Kurepa family in [A] °, the hypothesis of the second theorem is satisfied when the stationary cofinality of [A]*0 is equal to the size of A  ... 
doi:10.2307/2048422 fatcat:tjryqm2xerb2xbebxumld67iw4

Page 575 of Mathematical Reviews Vol. 24, Issue 6A [page]

1962 Mathematical Reviews  
For p, q, tr, and 8 infinite, the authors give a complete answer based on the generalized continuum hypothesis, and a partial answer without this hypothesis.  ...  The pre- sentation, which is sketchy, consists in listing of results and problems with hints to references. D. Kurepa (Zagreb) A3073 Cuesta, N. Mathematical generators.  ... 

Page 1763 of Mathematical Reviews Vol. 51, Issue 6 [page]

1976 Mathematical Reviews  
hypothesis is equivalent to weak compactness, and a generalised Kurepa hypothesis is equivalent to ineffability (a notion first found in this context).  ...  Logic 4 (1972), 229-308; erratum, ibid. 4 (1972), 443; MR 46 #8834], and the generalisations of the results on Suslin and Kurepa hypotheses to higher cardinals follow; thus if V=L, a generalised Suslin  ... 

Descriptive properties of higher Kurepa trees [article]

Philipp Lücke, Philipp Schlicht
2021 arXiv   pre-print
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  ...  of the generalized Baire space ^κκ of κ.  ...  The proof of the following theorem relies on a result of Donder on the structural properties of the Kurepa trees constructed from the canonical morasses at successor cardinals in the constructible universe  ... 
arXiv:2010.11597v2 fatcat:oxxrsdvmlzdunicg62vh7w3b4q

Kurepa trees and spectra of L_ω_1,ω-sentences [article]

Dima Sinapova, Ioannis Souldatos
2018 arXiv   pre-print
We construct a single 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 [_0,_ω_1] and also consistently equal to [_0,2  ...  Our result proves that 2^_0 can not be replaced by 2^_1, even if 2^_0<_ω_1.  ...  The ideas of Lemma 2.2 and Theorem 3.7 were communicated to the authors by Philipp Lücke. The consistency results of Section 3 were based on early ideas of Stevo Todorcevic.  ... 
arXiv:1705.05821v2 fatcat:pyyv6oiw5jf5jog3ld6xj4gxje

Page 1780 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews  
The generic Kurepa hypothesis (GKH) asserts that a Kurepa tree may be adjoined to the universe by ccc forcing.  ...  {See also the preceding review. } Dag Normann (N-OSLO) 92d:03037 03E35 03E45 03E55 Jensen, R. B. (4-OXAS); Schlechta, K. (D-IBM3) Results on the generic Kurepa hypothesis. Arch. Math.  ... 
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