{
  "abstract": "A famous conjecture of Erdős and Hajnal (1969) states that for every integer $g\\ge 4$ there is a smallest function $f_g:\\mathbb{N}\\to\\mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by Rödl (1977), with $f_4(k)$ bounded by a tower of height $Θ(k^2\\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\\ge 5$ there is a function $h_g:\\mathbb{N}\\to\\mathbb{N}$ growing as a power tower of height $\\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erdős and Hajnal (1975), and for $g=5$ improves Rödl's bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lovász-Theta-function of the complement), then for every $k,g\\in\\mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G'$ of odd-girth at least $g$ with $f(G')\\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\\ge 1$, let $\\mathcal{O}_p=\\{C_3,C_5,\\ldots,C_{2p+1}\\}$. We show that $R_k(\\mathcal{O}_p)\\ge(\\log^{(p-1)}k)^{k/3-o(k)}$ for every fixed $p$, where $\\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02522",
  "author_total": 1,
  "authors": [
    {
      "name": "Raphael Steiner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7js6p589q11f9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02522",
  "fulltext_url": null,
  "id": "7js6p589q11f9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02522",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7172402385",
  "pdf_url": "https://arxiv.org/pdf/2608.02522",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7js6p589q11f9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.02522",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02522"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Locally bipartite subgraphs via multicolor Ramsey numbers",
  "title_text": "Locally bipartite subgraphs via multicolor Ramsey numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/7js6p589q11f9",
  "venue": "arXiv",
  "versions": []
}