{
  "abstract": "Green's arithmetic regularity lemma over $G=\\F_2^n$ has tower-type lower bounds. We show that the large order properties in the Hosseini--Lovett--Moshkovitz--Shapira (HLM) construction survive sparse edits. If $d$ is the top block dimension, then changing at most $|G|/16$ points of the top HLM set $A_s$ still leaves a half-graph of height $d/8$. For a suitable choice of the HLM maps, changing at most $\\eps|G|$ points of a superlevel set of the averaged witness still leaves a half-graph of height $d/(8\\sqrt{s})\\ge{\\rm twr}(s-2)$. The proofs use Sauer's lemma and the binary code formed by the top-level row traces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19015",
  "author_total": 1,
  "authors": [
    {
      "name": "Dean Matthew Menezes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3k029edg89mfv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19015",
  "fulltext_url": null,
  "id": "3k029edg89mfv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19015",
  "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"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213498821",
  "pdf_url": "https://arxiv.org/pdf/2609.19015",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3k029edg89mfv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.19015",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19015"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Robust order properties in the HLM construction",
  "title_text": "Robust order properties in the HLM construction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/3k029edg89mfv",
  "venue": "arXiv",
  "versions": []
}