{
  "abstract": "We study the complementary version of Hedetniemi's problem for infinite graphs. We prove that if a graph $G$ and its complement $\\overline{G}$ are both uncountably chromatic while their categorical product is countably chromatic, then $|V(G)|=ω_1$. Assuming $\\diamondsuit$, we construct a graph $G$ on $ω_1$ such that $χ(G)=χ(\\overline{G})=ω_1$ and $χ(G\\times\\overline{G})=ω$; the construction uses two suitably chosen minimal Countryman lines. We also define a c.c.c. forcing of cardinality $ω_1$ that adds a graph with the same properties. It remains open whether ZFC alone proves the existence of such a graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08027",
  "author_total": 1,
  "authors": [
    {
      "name": "Lajos Soukup"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d83cb64jmkfer.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08027",
  "fulltext_url": null,
  "id": "d83cb64jmkfer",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08027",
  "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.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7220941489",
  "pdf_url": "https://arxiv.org/pdf/2610.08027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d83cb64jmkfer.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Hedetniemi's Conjecture for Uncountable Complementary Graphs",
  "title_text": "Hedetniemi's Conjecture for Uncountable Complementary Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/d83cb64jmkfer",
  "venue": "arXiv",
  "versions": []
}