# Tverberg cores and Kalai’s cascade conjecture

Authors: Pablo Soberón
Year: 2026
Status: submitted
Paper page: https://www.psoberon.com/papers/tverberg-cores/
arXiv: https://arxiv.org/abs/2605.21305

Topics: Tverberg theory; Topological 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.

## Overview

Topological Tverberg cores, vertex-deletion tolerance, and consequences for Kalai’s cascade conjecture.