A Galvin–Hajnal theorem for generalized cardinal characteristics

Lambie-Hanson, Chris

doi:10.1007/s40879-023-00610-7

Cited by 3 publications

(4 citation statements)

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“…The cardinal

\operatorname{m}(\kappa , \theta )

is more frequently found, especially the instances of it which follow from Shelah's Revised GCH ([29])—see for example, [1, Theorem 8.21] and [5, Theorem 8]. In [16] a Galvin–Hajnal theorem for

\operatorname{m}(\kappa , \theta )

is proved. In our notation we have followed [19] where the invariant

\operatorname{m}(\kappa , \theta )

was considered (though we order the parameters so as to follow Shelah's convention for the covering numbers from [27]).…”

Section: Preparatory Lemmatamentioning

confidence: 99%

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On strong chains of sets and functions

Inamdar

2023

Mathematika

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Shelah has shown that there are no chains of length ω3 increasing modulo finite in ω2ω2${}^{\omega _2}\omega _2$. We improve this result to sets. That is, we show that there are no chains of length ω3 in [ω2]ℵ2$[\omega _2]^{\aleph _2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2 increasing modulo finite in [ω1]ℵ1$[\omega _1]^{\aleph _1}$ as well as in ω1ω1${}^{\omega _1}\omega _1$. More generally, we study the depth of function spaces κμ${}^\kappa \mu$ quotiented by the ideal [κ]<θ$[\kappa ]^{< \theta }$ where θ<κ$\theta < \kappa$ are infinite cardinals.

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“…The cardinal

\operatorname{m}(\kappa , \theta )

is more frequently found, especially the instances of it which follow from Shelah's Revised GCH ([29])—see for example, [1, Theorem 8.21] and [5, Theorem 8]. In [16] a Galvin–Hajnal theorem for

\operatorname{m}(\kappa , \theta )

is proved. In our notation we have followed [19] where the invariant

\operatorname{m}(\kappa , \theta )

was considered (though we order the parameters so as to follow Shelah's convention for the covering numbers from [27]).…”

Section: Preparatory Lemmatamentioning

confidence: 99%

“…[5,Theorem 8]. In [16] a Galvin-Hajnal theorem for m(𝜅, 𝜃) is proved. In our notation we have followed [19] where the invariant m(𝜅, 𝜃) was considered (though we order the parameters so as to follow Shelah's convention for the covering numbers from [27]).…”

Section: Cardinal Invariantsmentioning

confidence: 99%

On strong chains of sets and functions

Inamdar

2023

Mathematika

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“…There has been an extensive research recently in the area of compactness principles at successor cardinals, and one of the questions is to what extent, if at all, these principles restrict the continuum function in the proximity of these cardinals. See for instance [5], [16], or [20] for some examples. Extending this question, we can ask whether there are some restrictions for other cardinal invariants besides the continuum function.…”

Section: Introductionmentioning

confidence: 99%

“…Extending this question, we can ask whether there are some restrictions for other cardinal invariants besides the continuum function. This has been done for instance in [15] and [13], where the focus is on the cardinal invariants on singular strong limit cardinals.…”

Section: Introductionmentioning

confidence: 99%

Small $$\mathfrak {u}(\kappa )$$ at singular $$\kappa $$ with compactness at $$\kappa ^{++}$$

Honzík

Stejskalová

2021

Arch. Math. Logic

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We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal κ and compactness principles at κ ++ . Let TP(λ), ¬wKH(λ), SR(λ) and DSS(λ) denote the tree property, the negation of the weak Kurepa Hypothesis, stationary reflection and the disjoint stationary sequence property, respectively, at a regular cardinal λ.We show that if the existence of a supercompact cardinal κ with a weakly compact cardinal λ above κ is consistent, then the following are consistent as well (where t(κ) and u(κ) are the tower number and the ultrafilter number, respectively):(i) There is an inaccessible cardinal κ such that κ + < t(κ) = u(κ) < 2 κ and SR(κ ++ ) and DSS(κ ++ ) hold, and(ii) There is an inaccessible cardinal κ such that κ + = t(κ) < u(κ) < 2 κ and SR(κ ++ ), DSS(κ ++ ), TP(κ ++ ) and ¬wKH(κ + ) hold.The cardinals u(κ) and 2 κ can have any reasonable values in these models. We obtain these results by combining the forcing construction from [4] due to Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results related to SR(λ), DSS(λ), TP(λ) and ¬wKH(λ). Apart from u(κ) and t(κ) we also compute the values of b(κ), d(κ), s(κ), r(κ), a(κ), cov(Mκ), add(Mκ), non(Mκ), cof(Mκ) which will all be equal to u(κ).In (ii), we compute p(κ) = t(κ) = κ + by observing that the κ +distributive quotient of the Mitchell forcing adds a tower of size κ + .Finally, as a corollary of the construction, we obtain that (i) and (ii) are also true for κ = ω (starting with a weakly compact cardinal in the ground model).

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On Compactness of Weak Square at Singulars of Uncountable Cofinality

LEVINE

2024

J. symb. log.

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Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at $\aleph _\omega $ , meaning that it is consistent that $\square _{\aleph _n}$ holds for all $n<\omega $ while $\square _{\aleph _\omega }$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild ${{\mathsf {PCF}}}$ -theoretic hypotheses, the weak square principle $\square _\kappa ^*$ is in fact compact at singulars of uncountable cofinality.

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A Galvin–Hajnal theorem for generalized cardinal characteristics

Abstract: We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin–Hajnal theorem, and hence also of Silver’s theorem, at singular cardinals of uncountable cofinality.

Cited by 3 publications

References 20 publications

On strong chains of sets and functions

On strong chains of sets and functions

Small $$\mathfrak {u}(\kappa )$$ at singular $$\kappa $$ with compactness at $$\kappa ^{++}$$

On Compactness of Weak Square at Singulars of Uncountable Cofinality

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