{
  "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 $2^{\\mathfrak{b}_κ} = 2^κ$. We also study the existence of $\\mathfrak{b}_κ$-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 $\\mathfrak{b}_κ$-scale set is a $K_κ$-space. Consequently, under this assumption, every $\\mathfrak{b}_κ$-scale set yields an alternative example of a $κ$-Menger space which is not a $K_κ$-space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33856",
  "author_total": 2,
  "authors": [
    {
      "name": "Nur Alam"
    },
    {
      "name": "Debraj Chandra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8r5yyb1s52k4v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33856",
  "fulltext_url": null,
  "id": "8r5yyb1s52k4v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33856",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.33856",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8r5yyb1s52k4v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33856",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33856"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Selection principles and the Scheepers diagram in an uncountable setting",
  "title_text": "Selection principles and the Scheepers diagram in an uncountable setting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/8r5yyb1s52k4v",
  "venue": "arXiv",
  "versions": []
}