{
  "abstract": "We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every $k\\geq 1$ and $α>0$, if $A$ is a subset of the leaves of $T_n$, where $T_n$ is the complete binary tree of height $n$, with size $|A|\\geq 2^{αn}$, then for $n$ sufficiently large the ancestor closed sub-tree $T_A\\subset T_n$ generated by $A$ must contain a copy of $T_k$ such that (1) all vertices in the same level of $T_k$ are mapped into vertices at the same level of $T_A$, (2) if a non-leaf vertex $x\\in T_k$ is mapped into a vertex $y \\in T_A$, then the two children of $x$ are mapped into descendants of the two children of $y$, and (3) the levels of $T_A$ occupied by the copy of $T_k$ form an arithmetic progression. For the product $T_n \\times T_n$ of two binary trees, we show that for every $k\\geq 1$ and $α> 1$, for $n$ sufficiently large, every subset $A$ of the leaves of $T_n\\times T_n$ of size at least $2^{αn}$ has that its ancestor closure $Γ_A\\subset T_n \\times T_n$ contains similarly structured arithmetic copies of the height $k$ four-ary tree, where the four children of every non-leaf vertex in the source tree are required to map below the four distinct \\textit{diagonal children} of the image vertex, where a diagonal child of a vertex $(x,y) \\in T_n \\times T_n$ is obtained by moving one level down each in each component. Our results generalise to product of $d$-many finite $b$-ary trees with a sharp critical exponent of \\[ α_{\\text{crit}}(d,b) =d-1 + \\log_b(b-1).\\] Geometrically, our results imply that any set $E\\subset [0,1)^d$ of upper Minkowski dimension greater than $α_{\\text{crit}}(d,b)$ contains arithmetic $b$-adic branching patterns of arbitrary finite order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28898",
  "author_total": 2,
  "authors": [
    {
      "name": "Alexander Fish"
    },
    {
      "name": "Sean Skinner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2t3d64v0y3hey.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28898",
  "fulltext_url": null,
  "id": "2t3d64v0y3hey",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28898",
  "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": "W7214399837",
  "pdf_url": "https://arxiv.org/pdf/2609.28898",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2t3d64v0y3hey.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28898",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28898"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ramsey Theory for Product Trees",
  "title_text": "Ramsey Theory for Product Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2t3d64v0y3hey",
  "venue": "arXiv",
  "versions": []
}