{
  "abstract": "We construct countable, stable, $ω$-categorical structures with at most two binary relations whose CSPs exhibit three complexity regimes: NP-complete, decidable but not in NP, and undecidable. The templates arise from a controlled Hrushovski construction of stable $ω$-categorical graphs $M_α$ for suitable irrational parameters $α>1$. Some of these graphs have NP-complete CSPs. Expanding the graphs $M_α$ by the non-edge relation turns age membership into a restriction of the CSP. For every time-constructible bound $t$, there is such an expansion whose CSP is decidable but not in DTIME$(t)$. For an explicit parameter $α$, the restriction to complete instances belongs to $\\mathrm{coNP}$ and the full CSP belongs to $Σ_2^{\\mathrm P}$. Separating the ages of the $M_α$ using graphs with large girth and prescribed density yields $2^{\\aleph_0}$ distinct CSPs of these expansions, all but countably many of which are undecidable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19940",
  "author_total": 4,
  "authors": [
    {
      "name": "Manuel Bodirsky"
    },
    {
      "name": "David Bradley-Williams"
    },
    {
      "name": "Ehud Hrushovski"
    },
    {
      "name": "Christoph Spiess"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ze7y142tshbh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19940",
  "fulltext_url": null,
  "id": "5ze7y142tshbh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19940",
  "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",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222039897",
  "pdf_url": "https://arxiv.org/pdf/2610.19940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ze7y142tshbh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19940"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Non-Dichotomy for CSPs of Stable $ω$-Categorical Structures",
  "title_text": "A Non-Dichotomy for CSPs of Stable ω-Categorical Structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/5ze7y142tshbh",
  "venue": "arXiv",
  "versions": []
}