  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.
