{
  "abstract": "Consider the linear form $L(x,y,z,w) = 3x+y-2z-2w$. For a positive integer $N$, denote by $r_L(N)$ the largest size of a subset of $\\{1,2,\\dots,N\\}$ that avoids nontrivial solutions to $L = 0$. We show that $r_L(N) = Ω(N^{0.5608687})$, improving the lower bound for Problem 16 in Green's list of open problems. Our proof uses the method of combinatorial degenerations to turn a finite certificate into large solution-free sets. In fact, we can improve Ruzsa's lower bound of $N^{1/2-o(1)}$ for many four-variable equations. Consider $L(x,y,z,w) = ax+by-cz-dw$ with $a,b,c,d\\in \\mathbb Z_{>0}$, $a+b=c+d$, $\\{a,b\\} \\neq \\{c,d\\}$ and $abcd$ not a square. We show that there is $\\varepsilon_L > 0$ such that $r_L(N) = Ω_L(N^{1/2 + \\varepsilon_L})$. Furthermore, we show that for primitive translation-invariant linear forms $L$ in $s$ variables, the lower bound $Ω_s(N^{1/(s-1)})$ coming from a greedy construction is never optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13972",
  "author_total": 2,
  "authors": [
    {
      "name": "Paul Hametner"
    },
    {
      "name": "Jozsef Solymosi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6wd1mkdqejp6d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13972",
  "fulltext_url": null,
  "id": "6wd1mkdqejp6d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13972",
  "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": "W7213232723",
  "pdf_url": "https://arxiv.org/pdf/2609.13972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6wd1mkdqejp6d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Large solution-free sets via combinatorial degenerations",
  "title_text": "Large solution-free sets via combinatorial degenerations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/6wd1mkdqejp6d",
  "venue": "arXiv",
  "versions": []
}