{
  "abstract": "We study weak and strong forbidden subposet problems in the linear lattice $L_n(q)$. We show that the $q$-analogues of the Bukh--Griggs--Lu conjecture fail for every prime power $q$. If $q\\ge 3$ and $n$ and $d$ have opposite parity, then $La_q(n,L_d(q))=La_q^*(n,L_d(q))=Σ_q(n,d)$. We also show that $La_q(n,D_s)=La_q^*(n,D_s)=Σ_q(n,2)$ for $2\\le s\\le q$ and all $n$, and for $s=q+1$ when $n$ is odd. For $n$ even, a maximum strong $D_{q+1}$-free family can be contained from the three middle layers. The case $s=2$ settles the $q$-analog of the diamond conjecture in the affirmative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37748",
  "author_total": 2,
  "authors": [
    {
      "name": "Balázs Patkós"
    },
    {
      "name": "Casey Tompkins"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12p2wb8yww5w2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37748",
  "fulltext_url": null,
  "id": "12p2wb8yww5w2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37748",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214997154",
  "pdf_url": "https://arxiv.org/pdf/2609.37748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12p2wb8yww5w2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Forbidden subposet problems in the linear lattice",
  "title_text": "Forbidden subposet problems in the linear lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/12p2wb8yww5w2",
  "venue": "arXiv",
  "versions": []
}