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PreprintCC BY-NC-SA 4.0arXiv

On colorful Helly numbers and the growth of Tverberg numbers

  • Chaya Keller
  • Skand Parvatikar
  • Tuan Tran
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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)2r, improving Holmsen--Lee's bound rrO(log r) and matching the known lower bound 2r−1−1 up to a factor O(r). Furthermore, we prove that the Tverberg number is rk=O(r⌈ log2 r⌉ k), reducing Pálvölgyi's coefficient rrrO(log r) to a quasipolynomial function of r. For S4-separable spaces, we prove rk=O(r2k). In the other direction, we construct a sequence of S4-separable spaces with Tverberg number rk=Ω(r2k). This shows that the weak Eckhoff conjecture, asserting that rk=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.

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Cite

BibTeXRIS
BibTeX
@misc{keller2026colorful,
  title = {{On colorful Helly numbers and the growth of Tverberg numbers}},
  author = {Chaya Keller and Skand Parvatikar and Tuan Tran},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.06437},
  eprint = {2608.06437},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.06437},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - On colorful Helly numbers and the growth of Tverberg numbers
AU  - Chaya Keller
AU  - Skand Parvatikar
AU  - Tuan Tran
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.06437
UR  - https://arxiv.org/abs/2608.06437
AB  - 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^⌈ log_2 r⌉ 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.
ER  -