{
  "abstract": "We determine the asymmetric homomorphism threshold from graphs of odd girth at least $7$ to triangle-free graphs, showing that $δ_{\\mathrm{hom}}(\\{C_3,C_5\\},\\{C_3\\})=\\frac{1}{9}$. Equivalently, for every $\\varepsilon>0$, every $n$-vertex graph of odd girth at least $7$ and minimum degree at least $(1/9+\\varepsilon)n$ admits a homomorphism to a triangle-free graph of size bounded by a function of $\\varepsilon$, while there exist graphs of odd girth at least $7$ and minimum degree at least $(1/9-\\varepsilon)n$ for which no such bounded-size triangle-free homomorphic image exists. More generally, for every $t\\geq 3$, we prove $δ_{\\mathrm{hom}}(\\{C_3,C_5\\},\\{K_t\\})=\\frac{1}{3t}$. In particular, for every proper monotone class $\\mathcal C$ of graphs, the threshold for graphs of odd girth at least $7$ to admit a homomorphism to a bounded-size graph in $\\mathcal C$ is positive. We further extend this phenomenon to arbitrary odd girth: for every $k\\geq 2$ and every proper monotone subclass $\\mathcal C$ of the class of graphs of odd girth at least $2k-1$, the threshold for graphs of odd girth at least $2k+3$ to admit a homomorphism to a bounded-size graph in $\\mathcal C$ is positive. These results disprove conjectures of Gishboliner, Hurley and Wigderson and exhibit a sharp contrast with the corresponding zero chromatic-threshold results. Our lower-bound constructions are based on high-dimensional Borsuk graphs, while the matching upper bounds use regularity arguments to recover the structure underlying these constructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25390",
  "author_total": 4,
  "authors": [
    {
      "name": "Romain Bourneuf"
    },
    {
      "name": "Raphael Steiner"
    },
    {
      "name": "Stéphan Thomassé"
    },
    {
      "name": "Yuval Wigderson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6qzawr0bxrvx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25390",
  "fulltext_url": null,
  "id": "d6qzawr0bxrvx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25390",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214103949",
  "pdf_url": "https://arxiv.org/pdf/2609.25390",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6qzawr0bxrvx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25390",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25390"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymmetric Homomorphism Thresholds for Graphs of Large Odd Girth",
  "title_text": "Asymmetric Homomorphism Thresholds for Graphs of Large Odd Girth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/d6qzawr0bxrvx",
  "venue": "arXiv",
  "versions": []
}