{
  "abstract": "Let $P$ be a fixed finite poset whose maximal chains all have $h$ elements, and let $\\mathcal W_{n,h}$ be the union of $h$ consecutive middle levels of $2^{[n]}$. We prove that the maximum size of a $P$-free subfamily of $\\mathcal W_{n,h}$ is $(π(P)+o(1))M_n$ for a constant $π(P)$, where $M_n=\\binom n{\\lfloor n/2\\rfloor}$, and that the number of $P$-free subfamilies is $2^{(π(P)+o(1))M_n}$. For connected $P$ we prove supersaturation above $(π(P)+ε)M_n$ and determine, in terms of the coordinate patterns of the copies of $P$ in the window, a range of $p$ in which the largest $P$-free subfamily of the random family $\\mathcal W_{n,h}(p)$ has size $(π(P)+o(1))pM_n$ and a range in which it has size $(h+o(1))pM_n$; the two ranges meet whenever all copies have the same pattern. For the Boolean poset $Q_d$, which has height $d+1$, the threshold order in the corresponding window $\\mathcal W_{n,d+1}$ is $n^{-d/(2^d-1)}$, and $π_d=π(Q_d)$ satisfies $(d+1)λ_d\\leπ_d\\le d+1-\\binom d{\\lfloor d/2\\rfloor}^{-1}$, where $λ_d$ is the vertex Turán density of $d$-dimensional subcubes in the hypercube. Together with a construction of Ellis, Ivan and Leader this gives $d+1-π_d=\\exp(-Θ(d))$. For the diamond, known bounds give $2.1479\\leπ_2\\le2.15121$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28413",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiangdong Ai"
    },
    {
      "name": "Mingyu Liu"
    },
    {
      "name": "Zhiyu Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dxwjf9kgs2wf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28413",
  "fulltext_url": null,
  "id": "1dxwjf9kgs2wf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28413",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223157509",
  "pdf_url": "https://arxiv.org/pdf/2610.28413",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dxwjf9kgs2wf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28413",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28413"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Forbidden graded posets in middle levels: supersaturation, enumeration and randomness",
  "title_text": "Forbidden graded posets in middle levels: supersaturation, enumeration and randomness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/1dxwjf9kgs2wf",
  "venue": "arXiv",
  "versions": []
}