{
  "abstract": "We study the removal problem for permutation patterns under the normalized $L_\\infty$ earth mover's distance. We prove a complete dichotomy. Let \\[ \\mathcal E =\\{(12),(21),(132),(213),(231),(312)\\}. \\] If $σ\\in\\mathcal E$, then every permutation of length $n$ that is $\\varepsilon$-far from being $σ$-free contains at least $\\varepsilon(n-1)$ copies of $σ$; for $0<\\varepsilon\\le 1/16$, this is sharp up to a constant factor for all sufficiently large $n$. If $σ\\in S_k\\setminus\\mathcal E$, then for every sufficiently large $n$ there is a permutation of length $n$ that is $1/(8k)$-far from being $σ$-free and contains exactly one copy of $σ$. Thus the usual positive-density removal conclusion fails for every nontrivial permutation pattern, but the sparse behavior is completely characterized: the minimum number of copies is linear for exactly six patterns and is one for all others. Consequently, no canonical one-sided tester that rejects only after finding a forbidden copy can have query complexity independent of the permutation length.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28271",
  "author_total": 3,
  "authors": [
    {
      "name": "Viet Hung Le"
    },
    {
      "name": "Ruilin Shi"
    },
    {
      "name": "Fan Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/862kdn24tyb16.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28271",
  "fulltext_url": null,
  "id": "862kdn24tyb16",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28271",
  "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": "W7223145418",
  "pdf_url": "https://arxiv.org/pdf/2610.28271",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/862kdn24tyb16.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28271",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28271"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Complete Dichotomy for $L_\\infty$ Permutation Removal",
  "title_text": "A Complete Dichotomy for L_∞ Permutation Removal",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/862kdn24tyb16",
  "venue": "arXiv",
  "versions": []
}