{
  "abstract": "For graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the minimum integer $N$ such that every $N$-vertex graph contains $F$ or its complement contains $H$. If $F$ is connected and $|F|\\geσ(H)$, a construction of Burr gives $R(F,H)\\ge(χ(H)-1)(|F|-1)+σ(H)$, where $σ(H)$ denotes the minimum order of a color class in a proper $χ(H)$-coloring of $H$. Burr proved that this bound is attained for $F=C_n$ when $n$ is sufficiently large. Allen, Brightwell and Skokan conjectured that equality already holds whenever $n\\geq |H| χ(H)$, while Haslegrave, Hyde, Kim and Liu subsequently proved it whenever $n\\ge C|H|\\log^4χ(H)$. Pokrovskiy and Sudakov conjectured that the optimal linear condition $n\\geq C|H|$ suffices; this conjecture was also highlighted by Montgomery in his 2026 ICM survey (see Conjecture 9.2). In this paper, we resolve this conjecture by proving that there is an absolute constant $C>0$ such that $R(C_n,H)=(χ(H)-1)(n-1)+σ(H)$ for every nonempty graph $H$ and every $n\\ge C|H|$. This gives the first bound linear in $|H|$, and is best possible up to a constant factor. Our proof builds on the framework of Haslegrave, Hyde, Kim and Liu, and combines some new ideas in expansion and switching cycle lengths.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.26956",
  "author_total": 2,
  "authors": [
    {
      "name": "Peiru Kuang"
    },
    {
      "name": "Yan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7f3wxjxx5yda8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.26956",
  "fulltext_url": null,
  "id": "7f3wxjxx5yda8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.26956",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7171856531",
  "pdf_url": "https://arxiv.org/pdf/2607.26956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7f3wxjxx5yda8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.26956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.26956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An Optimal Bound for Ramsey Goodness of Cycles",
  "title_text": "An Optimal Bound for Ramsey Goodness of Cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/7f3wxjxx5yda8",
  "venue": "arXiv",
  "versions": []
}