{
  "abstract": "For a graph $H$, let $f(n,e,H)$ be the least number of colors in an edge-coloring of some $n$-vertex graph with at least $e$ edges in which every copy of $H$ is rainbow. Burr, Erdős, Graham, and Sós conjectured that $f(n,\\lfloor n^2/4\\rfloor+1,C_{2k+1})=(1/8+o(1))n^2$ for every fixed $k\\ge3$, and Bucić, Chen, and Ma recently proved this for all $k\\ge4$. We prove the remaining case $k=3$: \\[ f\\left(n,\\left\\lfloor n^2/4\\right\\rfloor+1,C_7\\right) =\\left(\\frac18+o(1)\\right)n^2. \\] The lower bound rests on a weighted palette inequality, which we prove with an exact rational certificate on five sampled vertices. Its main ingredients are a fractional matching of compatible triangular edges and private resources attached to nontriangular edges. A stable form of the inequality, combined with regularity, triangle removal, and a direct argument for graphs close to bipartite, transfers the bound to arbitrary edge-colorings. We also describe a Lean 4 formalization of the conjecture for every fixed $k\\ge3$, which combines the new seven-cycle proof with a formalization of the Bucić-Chen-Ma argument for $k\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38286",
  "author_total": 1,
  "authors": [
    {
      "name": "Asad Shahab"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cqq5tqmv8ahvj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38286",
  "fulltext_url": null,
  "id": "cqq5tqmv8ahvj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38286",
  "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": "W7215075972",
  "pdf_url": "https://arxiv.org/pdf/2609.38286",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cqq5tqmv8ahvj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38286",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38286"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Burr-Erdős-Graham-Sós conjecture for the seven-cycle",
  "title_text": "The Burr-Erdős-Graham-Sós conjecture for the seven-cycle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/cqq5tqmv8ahvj",
  "venue": "arXiv",
  "versions": []
}