{
  "abstract": "We introduce a combinatorial principle $\\textsf{SC}(κ)$, for any regular cardinal $κ\\ge ω_2$, which asserts the existence of a family $\\mathcal{Y} \\subseteq [H(κ)]^ω$ of elementary substructures of $H(κ)$ which is closed under finite intersections and satisfies that for stationarily many $N \\in \\mathcal{Y}$, if $M \\in \\mathcal{Y}$ and $M \\cap ω_1 < N \\cap ω_1$, then $M \\cap N \\cap κ\\in N$. We explore the idea of defining forcings with side conditions consisting of finite subsets of such a family $\\mathcal{Y}$. If $\\textsf{SC}(ω_2)$ holds, then there exists a thin stationary subset of $[ω_2]^ω$, and consequently $\\textsf{SC}(ω_2)$ is independent of ZFC. Assuming that $μ$ is a Mahlo cardinal, the Lévy collapse $\\mathrm{Col}(ω,<μ)$ forces that $\\textsf{SC}(κ)$ holds for all regular cardinals $κ\\ge ω_2$. Local instances of $\\textsf{SC}$ can be forced using matrices of models or derived as consequences of square principles. We apply the principle $\\textsf{SC}$ to define two kinds of forcings: a streamlined version of a strongly proper collapse, and a forcing which is strongly proper on a stationary set and adds arbitrarily many Baumgartner clubs. Finally, we prove two consistency results using forcings derived from $\\textsf{SC}$ related to Aronszajn trees, the guessing model principle, strong failures of club guessing, and the higher dominating number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04362",
  "author_total": 2,
  "authors": [
    {
      "name": "Hannes Jakob"
    },
    {
      "name": "John Krueger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6n7bpbvpq3ex3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04362",
  "fulltext_url": null,
  "id": "6n7bpbvpq3ex3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04362",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04362",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6n7bpbvpq3ex3.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04362",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04362"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Side Condition Principle With Applications",
  "title_text": "A Side Condition Principle With Applications",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/6n7bpbvpq3ex3",
  "venue": "arXiv",
  "versions": []
}