Pablo SoberónMathematics · CUNY

Publications / 2026

2026 · submitted

Tverberg cores and Kalai’s cascade conjecture

Pablo Soberón

arXiv ↗PDF ↗BibTeX
Tverberg theoryTopological combinatorics

Abstract

We study topological analogues of Kalai's cascade conjecture. Given a continuous map from an n-simplex to ℝᵈ, let Tᵣ(f) be the set of points contained in the images of r pairwise disjoint faces. We prove that if r is a prime power and dim Tᵣ(f) ≤ k, then there exists a point that remains an r-Tverberg point after any t vertices are deleted, provided n = (r−1)(d+1) + t(k+1).

For t=1, this gives a topological analogue of a standard consequence of Kalai's cascade conjecture. We also confirm the cascade conjecture for finite point sets whose Radon set is 0-dimensional.

Illustration accompanying Tverberg cores and Kalai’s cascade conjecture
BibTeX citation

A minimal citation using the imported metadata. Journal references are retained in the note field.

@misc{soberon-2026-tverberg-cores,
  title = {{Tverberg cores and Kalai’s cascade conjecture}},
  author = {Pablo Soberón},
  year = {2026},
  eprint = {2605.21305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.21305}
}
Download .bib

Related papers