Selection principles and the Scheepers diagram in an uncountable setting

Abstract

We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the κ-Hurewicz Conjecture, and related questions on κ-Menger and K_κ-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact κ, we show that there is a model of ZFC in which the κ-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at κ. Moreover, any model in which κ is weakly compact and the κ-Hurewicz Conjecture holds must satisfy 2^b_κ = 2^κ. We also study the existence of b_κ-scale sets which are K_κ-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no b_κ-scale set is a K_κ-space. Consequently, under this assumption, every b_κ-scale set yields an alternative example of a κ-Menger space which is not a K_κ-space.

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BibTeX

@misc{alam2026selection,
  title = {{Selection principles and the Scheepers diagram in an uncountable setting}},
  author = {Nur Alam and Debraj Chandra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33856},
  eprint = {2609.33856},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33856},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Selection principles and the Scheepers diagram in an uncountable setting
AU  - Nur Alam
AU  - Debraj Chandra
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33856
UR  - https://arxiv.org/abs/2609.33856
AB  - We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the κ-Hurewicz Conjecture, and related questions on κ-Menger and K_κ-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact κ, we show that there is a model of ZFC in which the κ-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at κ. Moreover, any model in which κ is weakly compact and the κ-Hurewicz Conjecture holds must satisfy 2^b_κ = 2^κ. We also study the existence of b_κ-scale sets which are K_κ-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no b_κ-scale set is a K_κ-space. Consequently, under this assumption, every b_κ-scale set yields an alternative example of a κ-Menger space which is not a K_κ-space.
ER  -

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