A combinatorial approach to colourful simplicial depth

The colourful simplicial depth conjecture states that any point in the convex hull of each of d + 1 sets, or colours, of d + 1 points in general position in ℝd is contained in at least d2 + 1 simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral …