Computational Lower Bounds for Colourful Simplicial Depth

TL;DR

This paper investigates the colourful simplicial depth problem, aiming to improve lower bounds on the minimal number of colourful simplices containing the origin in high-dimensional configurations.

Contribution

It introduces a new method using invariants to analyze combinatorial octahedral systems, improving the lower bound for dimension 4.

Findings

Improved lower bound for d=4 to 9 simplices

Introduced invariants to exclude symmetric configurations

Enhanced understanding of colourful simplicial depth complexity

Abstract

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d^2+1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d=3, however the best known lower bound for d at least 4 is ((d+1)^2)/2. A promising method to improve this lower bound is to look at combinatorial octahedral systems generated by such configurations. The difficulty to employing this approach is handling the many symmetric configurations that arise. We propose a table of invariants which exclude many of partial configurations, and use this to improve the lower bound in dimension 4.