The topological Bárány–Larman conjecture for prime numbers

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.