A Note on Lower Bounds for Colourful Simplicial Depth

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, or 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 ≥ 4 is ？( d+1) 2 /2 ？. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.