  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 R^d
is contained in at least d^2+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 systems. We
present properties of octahedral systems generalizing earlier results on
colourful point configurations and exhibit an octahedral system which can not
arise from a colourful point configuration. The number of octahedral systems is
also given.
