{
  "abstract": "The size-Ramsey number $\\widehat{R}_k(\\mathcal G)$ of a $k$-uniform hypergraph $\\mathcal G$ is the minimum number of edges in a $k$-uniform hypergraph $\\mathcal H$ such that every $2$-edge-coloring of $\\mathcal H$ contains a monochromatic copy of $\\mathcal G$. Fox asked whether, for fixed $2\\le \\ell<k$, the size-Ramsey number of every $k$-uniform relaxed $\\ell$-tree on $n$ vertices is bounded by a polynomial in $n$, as recorded by Dudek, La Fleur, Mubayi and Rödl. We give a negative answer when $\\ell+1\\le k\\le 2\\ell-2$. More precisely, for every sufficiently large $n$, we construct a $k$-uniform relaxed $\\ell$-tree $\\bar{\\mathcal T}_{n,\\ell}^{(k)}$ on exactly $n$ vertices such that \\[ \\widehat{R}_k\\bigl(\\bar{\\mathcal T}_{n,\\ell}^{(k)}\\bigr) \\ge 2^{c_{k,\\ell}n^{\\,1-1/(2\\ell-k)}}, \\] where $c_{k,\\ell}>0$ depends only on $k$ and $\\ell$. Moreover, any two distinct edges of the construction intersect in at most $2\\ell-k-1$ vertices. In contrast, for $k\\ge 2\\ell$, we prove that every $k$-uniform relaxed $\\ell$-tree $\\bar{\\mathcal T}_{n,\\ell}^{(k)}$ on $n$ vertices satisfies \\[ \\widehat{R}_k\\bigl(\\bar{\\mathcal T}_{n,\\ell}^{(k)}\\bigr) =O_{k,\\ell}(n^{2k}). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04713",
  "author_total": 2,
  "authors": [
    {
      "name": "Meng Ji"
    },
    {
      "name": "Yulai Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cfxvwprq0kzqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04713",
  "fulltext_url": null,
  "id": "cfxvwprq0kzqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04713",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7211900258",
  "pdf_url": "https://arxiv.org/pdf/2609.04713",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cfxvwprq0kzqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.04713",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04713"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The size-Ramsey numbers of $k$-uniform relaxed $\\ell$-trees",
  "title_text": "The size-Ramsey numbers of k-uniform relaxed ℓ-trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/cfxvwprq0kzqe",
  "venue": "arXiv",
  "versions": []
}