The colourful simplicial depth conjecture

Given d+1 sets of points, or colours, S1,…,Sd+1 in Rd, a colourful simplex is a set T⊆⋃i=1d+1Si such that |T∩Si|≤1 for all i∈{1,…,d+1}. The colourful Carathéodory theorem states that, if 0 is in the convex hull of each Si, then there exists a colourful simplex T containing 0 in its convex hull. Deza et al. (2006) [3] conjectured that, when |Si|=d+1 for all i∈{1,…,d+1}, there are always at least d2+1 colourful simplices containing 0 in their convex hulls. We prove this conjecture via a combinatorial approach.