{
  "abstract": "A tree poset $T$ is a poset whose Hasse diagram is a tree. Bukh proved that if a family $\\mathcal F\\subseteq 2^{[n]}$ contains $(h(T)-1+\\varepsilon)\\binom{n}{\\lfloor \\frac{n}{2}\\rfloor}$ sets, then $\\mathcal F$ contains a weak copy of $T$, where $h(T)$ is the height of $T$, the number of elements in a longest chain of $T$. Several strengthenings and generalizations of this result have been obtained. We prove the following robust variant. For a tree poset $T$ and $x\\in T$, the $b$-blow-up $T(x,b)$ rooted at $x$ is the tree poset that we obtain from $T$ by replacing every element $u$ by $b^d$ new elements, where $d$ is the distance $d(x,u)$ in the Hasse diagram of $T$ and an edge $uv$ with $v$ being closer to $x$ is replaced by edges such that every new copy of $v$ is joined to $b$ new copies of $u$ such that these new copies form pairwise disjoint sets for the copies of $v$. We prove that for any tree poset $T$, $x\\in T$, and $\\varepsilon>0$ there exists $δ$ such that if $\\mathcal F\\subseteq 2^{[n]}$ contains $(h(T)-1+\\varepsilon)\\binom{n}{\\lfloor \\frac{n}{2}\\rfloor}$ sets, then $\\mathcal F$ contains a weak copy of $T(x,\\lfloor δn\\rfloor)$. This settles a conjecture of Treglown and the author. As applications, we derive the known asymptotic counting and random versions of Bukh's theorem from this stronger embedding result, and obtain new maximal anti-Ramsey results for tree posets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23543",
  "author_total": 1,
  "authors": [
    {
      "name": "Balázs Patkós"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn6s5er6s37r7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23543",
  "fulltext_url": null,
  "id": "bn6s5er6s37r7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23543",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214058690",
  "pdf_url": "https://arxiv.org/pdf/2609.23543",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn6s5er6s37r7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23543",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23543"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Embedding rooted blow-ups of tree posets",
  "title_text": "Embedding rooted blow-ups of tree posets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bn6s5er6s37r7",
  "venue": "arXiv",
  "versions": []
}