# The topological Bárány–Larman conjecture for prime numbers

Authors: Pablo Soberón
Year: 2026
Status: preprint
Paper page: https://www.psoberon.com/papers/barany-larman-primes/
arXiv: https://arxiv.org/abs/2609.37876

Topics: Tverberg theory; Topological combinatorics

## Abstract

The Bárány–Larman conjecture states that for any d+1 sets of r points each in ℝᵈ, considered as color classes, we can partition their union into r rainbow (d+1)-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when r is a prime number. We also show that the optimal colorful Tverberg theorem of Blagojević, Matschke, and Ziegler cannot be extended to prime powers.

## Overview

The topological Bárány–Larman conjecture for prime numbers, and counterexamples to extending the optimal colorful Tverberg theorem to prime powers.