{
  "abstract": "We study fractional coloring with local demands, a framework introduced by Kelly and Postle in which one seeks a probability distribution on the independent sets of a graph such that each vertex $v$ is included with probability at least a prescribed target $ϕ(v)$. For $K_r$-free graphs, Kelly and Postle conjectured that such a distribution exists provided $ϕ(v)\\le c_r\\frac{\\log d(v)}{d(v)}$ for a fixed \\(c_r>0\\), where $d(v)$ is the degree of $v$, and they established this up to a factor of $(\\log\\log d(v))^2$. In this paper, we prove their conjecture, showing that \\[ ϕ(v)\\le\\frac{c}{r}\\frac{\\log(d(v)+2)}{d(v)+1} \\] suffices for an absolute constant $c>0$. As a corollary, we obtain a lower bound on the independence number of $K_r$-free graphs in terms of their degree sequence. This strengthens the bound that OpenAI proved in resolving the Ajtai-Erdős-Komlós-Szemerédi conjecture. Moreover, our argument yields a deterministic algorithm that produces the desired probability distribution in time $2^{O(r^2)}n^{O(1)}$, where \\(n\\) is the number of vertices and the implied constants are absolute.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04297",
  "author_total": 5,
  "authors": [
    {
      "name": "Ewan Davies"
    },
    {
      "name": "Abhishek Dhawan"
    },
    {
      "name": "Abhishek Methuku"
    },
    {
      "name": "Huy Nguyen"
    },
    {
      "name": "Jaehyeon Seo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1mjyrqhqn75k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04297",
  "fulltext_url": null,
  "id": "b1mjyrqhqn75k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04297",
  "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.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04297",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1mjyrqhqn75k.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04297",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04297"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fractional colorings of $K_r$-free graphs with local demands",
  "title_text": "Fractional colorings of K_r-free graphs with local demands",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/b1mjyrqhqn75k",
  "venue": "arXiv",
  "versions": []
}