{
  "abstract": "We establish new bounds on colorful Helly and Tverberg numbers in abstract convexity spaces with Radon number $r$. We show that the colorful Helly number is at most $(r-1)2^r$, improving Holmsen--Lee's bound $r^{r^{O(\\log r)}}$ and matching the known lower bound $2^{r-1}-1$ up to a factor $O(r)$. Furthermore, we prove that the Tverberg number is $r_k=O(r^{\\lceil \\log_2 r\\rceil }k)$, reducing Pálvölgyi's coefficient $r^{r^{r^{O(\\log r)}}}$ to a quasipolynomial function of $r$. For $S_4$-separable spaces, we prove $r_k=O(r^2k)$. In the other direction, we construct a sequence of $S_4$-separable spaces with Tverberg number $r_k=Ω(r^2k)$. This shows that the weak Eckhoff conjecture, asserting that $r_k=O(rk)$ in any abstract convexity space, fails already in separable spaces. The same construction refutes the analogue of Reay's conjecture in separable spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06437",
  "author_total": 3,
  "authors": [
    {
      "name": "Chaya Keller"
    },
    {
      "name": "Skand Parvatikar"
    },
    {
      "name": "Tuan Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b5znbnnwkfat6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06437",
  "fulltext_url": null,
  "id": "b5znbnnwkfat6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06437",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7202123714",
  "pdf_url": "https://arxiv.org/pdf/2608.06437",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b5znbnnwkfat6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2608.06437",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06437"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On colorful Helly numbers and the growth of Tverberg numbers",
  "title_text": "On colorful Helly numbers and the growth of Tverberg numbers",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/b5znbnnwkfat6",
  "venue": "arXiv",
  "versions": []
}