{
  "abstract": "For a graph $G$ with adjacency matrix $A_G$ and a coefficient field $\\mathbb K$, put $ρ_{\\mathbb K}(G)=\\operatorname{rank}_{\\mathbb K}(A_G+I)$. We answer the triangle-free Rank--Ramsey existence question of Beniamini, Linial and Shraibman. For every order $n$, we construct a triangle-free graph $H_n$ such that, simultaneously over all coefficient fields, $$ ρ_{\\mathbb K}(H_n)=\\bigl(4(2/3)^{3/4}+o(1)\\bigr)n^{3/4}, $$ with an error uniform in the field. The graphs can be chosen Hamiltonian for all sufficiently large $n$. Our method combines ordered layers of point--line incidence flags with a common low-dimensional endpoint space. We determine the exact real rank, independence number and chromatic number of the affine lifts. For all triangle-free complete incidence lifts, we obtain a sharp two-term lower bound for complete incidence lifts, with sharpness witnessed by projective-plane incidence graphs over suitable coefficient fields. We also establish a third-order estimate and stability near equality. On the same explicit graphs, both the Lovász number and minimum semidefinite rank exceed the prescribed Boolean-matrix rank by polynomial factors. Thus neither parameter has a universal constant-factor upper bound in terms of complement rank. We determine Shannon capacity in the short-layer range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22902",
  "author_total": 3,
  "authors": [
    {
      "name": "Yong Lu"
    },
    {
      "name": "Qi Wu"
    },
    {
      "name": "Qiannan Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2x4ymmjny0ak1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22902",
  "fulltext_url": null,
  "id": "2x4ymmjny0ak1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22902",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7222929247",
  "pdf_url": "https://arxiv.org/pdf/2610.22902",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2x4ymmjny0ak1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22902",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22902"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Triangle-free Rank--Ramsey graphs from ordered incidence lifts",
  "title_text": "Triangle-free Rank--Ramsey graphs from ordered incidence lifts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/2x4ymmjny0ak1",
  "venue": "arXiv",
  "versions": []
}