Title:  A Combinatorial Approach to the Colourful Simplicial Depth

Given 3 blue points, 3 red points, and 3 green points in the plane so that the convex hull of each of those 3 mono-chromatic sets contains the origin, then it is possible to find a blue point, a red point, and a green point forming a triangle containing the origin. More generally, in dimension d, consider d+1 sets S1, S2,�,Sd+1 of d+1 points so that the convex hull of each Si contains the origin. The Colourful Carath�odory Theorem proven by B�r�ny in 1982 states that under these conditions, you can always choose a point in each Si forming the vertices of a colourful simplex containing the origin in its convex hull. We are interested in determining the minimum number μ(d) of colourful simplices that contain the origin over all sets satisfying the conditions of the Colourful Carath�odory Theorem. We present currently known properties of μ(d) which is conjectured to be equal to d2+1, and investigate a combinatorial generalization where a colourful simplex containing the origin corresponds to a hyper-edge in a hyper-graph, and B�r�ny's geometric conditions correspond to purely combinatorial conditions. In this talk we present new combinatorial insights and present preliminary results towards an improved lower bound.