{
  "abstract": "We prove a general upper bound for scaling windows of sparse monotone covering problems. From that, we deduce that for every fixed value $k\\geq 3$, the window of random $k$-SAT is $O(n/\\log n)$, improving the Friedgut-Bourgain bound of $O(n/\\log\\log n)$. We also show that random signed Not-All-Equal-$k$-SAT and hypergraph non-two-colourability (Property B) have windows of the same order. Our upper bound for scaling windows of sparse monotone covering problems follows from a general strategy combining the strengthening of Bourgain's sharp threshold theorem by Keevash, Lifshitz, Long, and Minzer with a local-to-random replacement principle. This provides a systematic method for proving scaling-window bounds, addressing a question of Perkins.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26222",
  "author_total": 1,
  "authors": [
    {
      "name": "Gaia Carenini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a190emqcervw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26222",
  "fulltext_url": null,
  "id": "2a190emqcervw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26222",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214078447",
  "pdf_url": "https://arxiv.org/pdf/2609.26222",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2a190emqcervw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26222",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26222"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Scaling Window of Random $k$-SAT",
  "title_text": "The Scaling Window of Random k-SAT",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2a190emqcervw",
  "venue": "arXiv",
  "versions": []
}