{
  "abstract": "A graph $G$ is $n$-existentially closed or $n$-e.c. if, for all subsets $S\\subseteq V(G)$ with $|S|=n$ and for all partitions $S=A\\sqcup B$, there exists a vertex in $V(G)\\setminus S$ adjacent to all vertices in $A$ and no vertices in $B$. We study the minimum number of edges $m(v,n)$ of a $v$-vertex $n$-e.c. graph, and show that $m(v,2)=3v+O(1)$ while $m(v,n)=Θ(v\\log v)$ for fixed $n\\ge 3$. The latter result uses a connection to binary covering arrays. A related parameter is the VC-dimension of $G$, defined as the size of the largest subset of vertices shattered by the neighborhoods of vertices in $G$. We initiate systematic study of the VC-dimensions of strongly regular graphs (SRGs). We characterize the sufficiently large SRGs with VC-dimension 2. Furthermore, we determine the VC-dimension of sufficiently large Latin square graphs and of all SRGs of order at most 28, and prove upper bounds for SRGs arising from orthogonal arrays or Steiner systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10330",
  "author_total": 3,
  "authors": [
    {
      "name": "Isabel Byrne"
    },
    {
      "name": "John Byrne"
    },
    {
      "name": "Sebastian M. Cioabă"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2rgdb5yze8485.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10330",
  "fulltext_url": null,
  "id": "2rgdb5yze8485",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10330",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212134217",
  "pdf_url": "https://arxiv.org/pdf/2609.10330",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2rgdb5yze8485.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2609.10330",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10330"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The VC-dimension of strongly regular graphs",
  "title_text": "The VC-dimension of strongly regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2rgdb5yze8485",
  "venue": "arXiv",
  "versions": []
}