{
  "abstract": "We study affine arrangements of an even number $n$ of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to $\\lfloor n(n-5/2)/3\\rfloor$. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of $n=6\\cdot2^t+2$ ($8,14,26,50,\\ldots$) straight lines with two triple points and $(n(n-5/2)+1)/3$ triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even $n\\le12$. For $n=6,8,10,12,14$ the maximum number of triangles, for both lines and pseudolines, is $7,15,25,38,54$; for $n=8,12,14$ every optimal arrangement has multiple points. All optimal arrangements for even $n\\le12$ are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value $n=11$, no arrangement with $33$ triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For $n\\equiv2\\pmod 6$ this again gives one triangle more than the simple bound, in pseudolines up to $n=44$ and in straight lines for $n=20$ and $n=32$. Doubling these pseudoline arrangements gives members of the series $18\\cdot2^t+2$ and $30\\cdot2^t+2$ with the same excess, up to $n=146$ and $n=242$, respectively; those with $n=38$ and $n=74$ are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number $n$ of pseudolines has at most $\\lfloor n(n-2)/3\\rfloor-\\lfloor n/6\\rfloor$ triangles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03096",
  "author_total": 2,
  "authors": [
    {
      "name": "Roman Parpalak"
    },
    {
      "name": "Denis Utkin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3r0jf5wg1f72w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03096",
  "fulltext_url": null,
  "id": "3r0jf5wg1f72w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03096",
  "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",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03096",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3r0jf5wg1f72w.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03096",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03096"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Non-simple Arrangements of Lines and Pseudolines with Many Triangles",
  "title_text": "Non-simple Arrangements of Lines and Pseudolines with Many Triangles",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/3r0jf5wg1f72w",
  "venue": "arXiv",
  "versions": []
}