The generalized Kurepa hypothesis at singular cardinals
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The generalized Kurepa hypothesis at singular cardinals Mohammad Golshani1
© Akadémiai Kiadó, Budapest, Hungary 2018
Abstract We discuss the generalized Kurepa hypothesis KHλ at singular cardinals λ. In particular, we answer questions of Erd˝os–Hajnal (in: Proceedings of the Symposium Pure Mathematics, Part I, University of California, Los Angeles, CA, 1967) and Todorcevic (Trees and linearly ordered sets. Handbook of set-theoretic topology, North-Holland, Amsterdam, pp. 235–293, 1984, Israel J Math 52(1–2): pp. 53–58, 1985) by showing that GCH does not imply KHℵω nor the existence of a family F ⊆ [ℵω ]ℵ0 of size ℵω+1 such that F X has size ℵ0 for every X ⊆ S, |X | = ℵ0 . Keywords Generalized Kurepa hypothesis · Singular cardinals · Chang’s conjecture · GCH
1 Introduction For an infinite cardinal λ let the generalized Kurepa hypothesis at λ, denoted KHλ , be the assertion: there exists a family F ⊆ P(λ) such that |F | > λ but |F X | ≤ |X | for every infinite X ⊆ λ, |X | < λ, where F X = {t ∩ X : t ∈ F }. By a theorem of Erd˝os–Hajnal–Milner [2], if λ is a singular cardinal of uncountable cofinality, θ cf(λ) < λ for all θ < λ and if F ⊆ P(λ) is such that the set {α < λ : |F α| ≤ |α|} is stationary in λ, then |F | ≤ λ. In particular, GCH implies KHλ fails for all singular cardinals λ of uncountable cofinality. On the other hand, by an unpublished result of Prikry [5], KHλ holds in L, the Gödel’s constructible universe, for singular cardinals of countable cofinality (see [7]). Later, Todorcevic [6,7] improved Prikry’s theorem by showing that if λ is a singular cardinal of countable cofinality, then λ implies KHλ . The following question is asked in [6] and [7]. Question 1.1 Does GCH imply KHℵω ?
The author’s research has been supported by a grant from IPM (No. 97030417). He also thanks Menachem Magidor for sharing his ideas during the 14th International Workshop in Set Theory at Luminy, in particular, the results of this paper are obtained from discussions with him.
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Mohammad Golshani [email protected] http://math.ipm.ac.ir/golshani/ School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
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The question is also related to the following question of Erd˝os–Hajnal [1] (question 19/E) Question 1.2 Assume GCH. Let |S| = ℵω . Does there exist a family F , |F | = ℵω+1 , F ⊆ [S]ℵ0 such that F X has size ℵ0 for every X ⊆ S, |X | = ℵ0 ? We show that, relative to the existence of large cardinals, both of the above questions can consistently be false, and so they are independent of ZFC.
2 KH fails above a supercompact cardinal In this section we prove the following. Theorem 2.1 Suppose κ is a supercompact cardinal and λ ≥ κ. Then KHλ fails. Proof Let F ⊆ P(λ) be of size ≥ λ+ . Let j : V → M be a λ+ -supercompactness embedding with crit( j) = κ. Also let U be the normal measure on Pκ (λ) derived from j, i.e., U = {X ⊆ Pκ (λ) : j[λ] ∈ j(X )}. We have • • • •
M | “ j(F ) ⊆ P( j(λ)) is of size ≥ j(λ)+ ”. j
[λ] ∈ M and M | “| j
[λ
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