{
  "abstract": "In 1993, Ruzsa showed that for every $k \\geq 2$, there exists a constant $C$ such that every subset $A \\subseteq [N]$ of size at least $C N^{1/k}$ contains $2k$ distinct elements $a_1, \\ldots, a_k, b_1, \\ldots, b_k \\in A$ such that $a_1 + \\cdots + a_k = b_1 + \\cdots + b_k$. We strengthen this result by proving that the elements $a_1, \\ldots, a_k, b_1, \\ldots, b_k$ can be chosen to have the additional property that $\\{a_1, \\ldots, a_k, b_1, \\ldots, b_k\\}$ has $2^{2k}-1$ distinct subset sums, with the only coincidence being that $\\{a_1, \\ldots, a_k\\}$ and $\\{b_1, \\ldots, b_k\\}$ have the same sum. Our proof also applies to any finite Abelian group of odd order $N$, and it provides a corresponding supersaturation result: that whenever $|A| \\geq CN^{1/k}$, there are at least $Ω(|A|^{2k}/N)$ choices for $a_1, \\ldots, a_k, b_1, \\ldots, b_k \\in A$ satisfying these properties. We prove a slightly weaker statement for Abelian groups of even order. We also apply our methods to the vector space setting and prove the following $\\mathbb F_q$-analogue of the Bondy-Simonovits Theorem on the extremal number of even cycles in graphs: Any rank-$n$, simple, $\\mathbb F_q$-representable matroid with no circuit of size exactly $2k$ has size at most $C q^{n/k}$ for some constant $C$ depending only on $q$ and $k$. When $q=2$, this is best possible up to the constant $C$ for all $k \\geq 2$. Our methods also apply to the original graph setting and give a new proof of the Bondy-Simonovits Theorem and its supersaturation version.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02177",
  "author_total": 2,
  "authors": [
    {
      "name": "Bryce Frederickson"
    },
    {
      "name": "Liana Yepremyan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7r2qvwh16pv9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02177",
  "fulltext_url": null,
  "id": "7r2qvwh16pv9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02177",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215810055",
  "pdf_url": "https://arxiv.org/pdf/2610.02177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7r2qvwh16pv9k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.02177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generic solutions to symmetric linear equations",
  "title_text": "Generic solutions to symmetric linear equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/7r2qvwh16pv9k",
  "venue": "arXiv",
  "versions": []
}