{
  "abstract": "For a finite set $A$ of positive integers, let $H(A)$ be its set of subset sums, and let $g_3(n)$ be the least $N$ for which some $n$-element set $A\\subseteq\\{1,\\ldots,N\\}$ has $H(A)$ free of nonconstant three-term arithmetic progressions. Erdős and Sárkőzy asked whether $g_3(n)\\gg 3^n$. We prove \\[ \\liminf_{n\\to\\infty}\\frac{g_3(n)}{3^n}=0. \\] More precisely, for every $ε>0$ there is an integer $d\\ge2$ such that $g_3(d\\ell)\\le ε3^{d\\ell}$ for all sufficiently large $\\ell$. The proof uses Korsky's characterization of the problem in terms of ternary coefficient sums and a consequence of an OpenAI construction that provides positive integer coefficients whose linear form is injective on large integer boxes. A base-three expansion then gives the result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06303",
  "author_total": 1,
  "authors": [
    {
      "name": "Simone Costa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2s3e1w3cg4gf6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06303",
  "fulltext_url": null,
  "id": "2s3e1w3cg4gf6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06303",
  "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": "W7212027896",
  "pdf_url": "https://arxiv.org/pdf/2609.06303",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2s3e1w3cg4gf6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06303",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06303"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A negative answer to the Erdős-Sárkőzy question",
  "title_text": "A negative answer to the Erdős-Sárkőzy question",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2s3e1w3cg4gf6",
  "venue": "arXiv",
  "versions": []
}