{
  "abstract": "Kahn proved that, among bipartite $d$-regular graphs on $n$ vertices, the number of independent sets is maximized by a disjoint union of copies of $K_{d,d}$. Zhao later extended this result to all $d$-regular graphs. We prove a robust version of this theorem in which independent sets are replaced by sets spanning few internal edges. If $G$ is $d$-regular on $n$ vertices, then the number of subsets spanning at most $γdn$ edges is at most $$ 2^{n/2}\\exp\\left\\{ O\\bigl(γ\\log(e/γ)n\\bigr) + O\\bigl(n/d\\bigr) \\right\\}. $$ Both correction terms are sharp up to absolute constants: the $γ\\log(1/γ)n$ term is necessary when $d$ is sufficiently large in terms of $γ$, while the $n/d$ term is already necessary for independent sets. Our result answers a question of Seth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28527",
  "author_total": 1,
  "authors": [
    {
      "name": "Gaia Carenini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2z0815w062bc7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28527",
  "fulltext_url": null,
  "id": "2z0815w062bc7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28527",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214402180",
  "pdf_url": "https://arxiv.org/pdf/2609.28527",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2z0815w062bc7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28527",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28527"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting almost independent sets in regular graphs",
  "title_text": "Counting almost independent sets in regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2z0815w062bc7",
  "venue": "arXiv",
  "versions": []
}