{
  "abstract": "Choi, Erdős and Szemerédi showed that there exists an absolute constant $C$ such that for all subsets $A \\subseteq \\{1, 2, \\ldots, 2n\\}$ with at least $n+C$ elements, there exist four distinct positive integers whose pairwise sums are all contained in $A$. A proof that one can take $C = 3166$ was recently sketched by the first author, and here we show that we actually have $C = 4$ for all $n \\ge 6$, which is optimal. The proof we present was originally conceived of by AI, with the final result proved and formalized by Aristotle, the formal reasoning agent developed by Harmonic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39314",
  "author_total": 2,
  "authors": [
    {
      "name": "Wouter van Doorn"
    },
    {
      "name": "John Erlbacher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2215ywnsmv5g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39314",
  "fulltext_url": null,
  "id": "c2215ywnsmv5g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39314",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215029362",
  "pdf_url": "https://arxiv.org/pdf/2609.39314",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2215ywnsmv5g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39314",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39314"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The cardinality of a set containing the pairwise sums of four positive integers",
  "title_text": "The cardinality of a set containing the pairwise sums of four positive integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/c2215ywnsmv5g",
  "venue": "arXiv",
  "versions": []
}