{
  "abstract": "Two distinct $r$-element sets are adjacent if their intersection has size $r-1$. The Kleitman-West problem, equivalently formulated as the edge-isoperimetric problem on the Johnson graph $J(n,r)$, asks for the maximum number $M_r(n,m)$ of unordered adjacent pairs in a family of $m$ $r$-element subsets of an $n$-element set, and for the families attaining this maximum. For every fixed integer $r\\ge2$, we determine this maximum asymptotically as $n\\to\\infty$, uniformly over the entire range of family sizes. Specifically, for each integer $1\\le j\\le r$, define $F_j(p)=\\max\\{p^{1+1/j},\\,2p-1+(1-p)^{1+1/j}\\}$ for $0\\le p\\le1$, and $G_j(x)=\\lfloor x\\rfloor+F_j(x-\\lfloor x\\rfloor)$ for $x\\ge0$. With \\[ P_r(n,m)=\\sum_{j=1}^r\\frac{n^{j+1}}{2(j-1)!}G_j\\!\\left(\\frac{j!m}{n^j}\\right)-\\frac{r(r-1)}4mn-\\frac m2, \\] we prove that \\[ \\lim_{n\\to\\infty}\\max_{2\\le m\\le\\binom nr}\\left|\\frac{M_r(n,m)}{P_r(n,m)}-1\\right|=0. \\] The remaining cases satisfy $M_r(n,0)=M_r(n,1)=0$. The proof first establishes the asymptotic formula for dense families using a sharp inequality for bounded random variables, with equality attained by a random variable taking at most two values. To extend the result uniformly to all family sizes, we extract groups of sets sharing a fixed subset, apply the dense estimates after deleting that subset, and control the adjacent pairs not contained in any one group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05162",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiasheng Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/201hmp3kfr8d9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05162",
  "fulltext_url": null,
  "id": "201hmp3kfr8d9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05162",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/201hmp3kfr8d9.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Uniform asymptotics for the Kleitman-West problem",
  "title_text": "Uniform asymptotics for the Kleitman-West problem",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/201hmp3kfr8d9",
  "venue": "arXiv",
  "versions": []
}