{
  "abstract": "The diameter graph of a finite set $S\\subset\\mathbb{R}^d$ has vertex set $S$ and two points are adjacent when their distance equals $\\operatorname{diam} S$. In this paper we study the following question: given $k$ and $d$, is there a constant $c=c(k,d)$ such that from every finite point set $S\\subset\\mathbb{R}^d$ one can remove at most $c$ points so that the subgraph of the diameter graph induced by the remaining points is $k$-colorable? Brass conjectured that such a constant exists for $d=3$ and $k=3$. We prove this conjecture by showing that at most one point needs to be removed. We also answer a question of Dol'nikov, showing that for every positive integer $m$, there exist a finite point set $S\\subset\\mathbb{R}^3$ in which every subset of smaller diameter has less than $(|S|-m)/2$ points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05631",
  "author_total": 1,
  "authors": [
    {
      "name": "Gyivan Lopez-Campos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23my2khcvzkyf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05631",
  "fulltext_url": null,
  "id": "23my2khcvzkyf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05631",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05631",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23my2khcvzkyf.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05631",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05631"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Diameter Graphs Problems: A proof of Brass's Conjecture",
  "title_text": "On Diameter Graphs Problems: A proof of Brass's Conjecture",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/23my2khcvzkyf",
  "venue": "arXiv",
  "versions": []
}