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Splitting positive sets

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Abstract

We introduce a class of cardinal invariants \({\mathfrak{s}_{\cal I}}\) for ideals \({\cal I}\) on ω which arise naturally from the FinBW property introduced by Filipów et al. (2007). Let \({\cal I}\) be an ideal on ω. Define

$${\mathfrak{s}_{\cal I}} = \min \{ |X|:X \subset {[\omega ]^\omega },\forall B \in {{\cal I}^ + },\,\exists x \in X(B\backslash x,B \cap x \in {[\omega ]^\omega })\}.$$

We characterize them and compare them with other cardinal invariants of the continuum.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601443, 11801386 and 11771311). The authors are grateful to the referees for careful reading of the manuscript, corrections, and comments which greatly improved this paper. Theorem 4.1 was proved only for the case where λ = ω1 and κ = ω2 in an early version of this paper. Both referees encourage us to prove the generalized version. One of the referees kindly provide (3) and (4) of Lemma 4.4. He (or She) also provides Remark 4.11.

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 610756, China

    Hang Zhang

  2. College of Mathematics, Sichuan University, Chengdu, 610064, China

    Jialiang He & Shuguo Zhang

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  1. Hang Zhang
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  2. Jialiang He
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  3. Shuguo Zhang
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Correspondence to Hang Zhang.

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Zhang, H., He, J. & Zhang, S. Splitting positive sets. Sci. China Math. 66, 2457–2470 (2023). https://doi.org/10.1007/s11425-022-2066-x

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  • DOI: https://doi.org/10.1007/s11425-022-2066-x

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