{
  "abstract": "We give a common counterexample to two product-structure conjectures in extremal graph theory. More precisely, we construct a fixed nonempty finite family $\\mathcal L$ with $p(\\mathcal L)=2$ such that, for some $c>0$, \\[ \\operatorname{ex}(n,\\mathcal L)>t_2(n)+cn^{3/2} \\] for every sufficiently large $n$. Nevertheless, at every such order there is an $\\mathcal L$-extremal graph whose complement is connected and which therefore admits no decomposition as a join of two nonempty graphs. This superlinear surplus also forces the decomposition family of $\\mathcal L$ to contain no forest. The construction uses endpoint-injective repair with a finite obstruction family admitting an exact extremal formula and equality classification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02115",
  "author_total": 1,
  "authors": [
    {
      "name": "Chuandong Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x8xgerphere5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02115",
  "fulltext_url": null,
  "id": "0x8xgerphere5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02115",
  "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": "W7172411634",
  "pdf_url": "https://arxiv.org/pdf/2608.02115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x8xgerphere5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.02115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A finite forbidden family with superlinear surplus and non-join extremal graphs",
  "title_text": "A finite forbidden family with superlinear surplus and non-join extremal graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/0x8xgerphere5",
  "venue": "arXiv",
  "versions": []
}