{
  "abstract": "A finite set of positive integers is admissible if any two of its subsets with the same sum have the same cardinality. In 1991, Erdős, Nicolas and Sárközy conjectured that $\\log a(N)\\asymp\\sqrt N$, where $a(N)$ denotes the number of nonempty admissible subsets of $\\{1,\\ldots,N\\}$. We prove this conjecture by showing that there are absolute constants $c,C>0$ such that $\\exp(c\\sqrt N)\\le a(N)\\le\\exp(C\\sqrt N)$ for every integer $N\\ge4$. The same bounds hold for the number of admissible sets with maximum $N$. For the upper bound, we show that every sufficiently large admissible set $A\\subseteq\\{1,\\ldots,N\\}$ contains a subset of proportional size whose doubling constant is $O(N/|A|^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34372",
  "author_total": 2,
  "authors": [
    {
      "name": "Guoqing Wang"
    },
    {
      "name": "Wenkai Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0q58b5fma7j33.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34372",
  "fulltext_url": null,
  "id": "0q58b5fma7j33",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34372",
  "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": "W7223888931",
  "pdf_url": "https://arxiv.org/pdf/2610.34372",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0q58b5fma7j33.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34372",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34372"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture",
  "title_text": "Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0q58b5fma7j33",
  "venue": "arXiv",
  "versions": []
}