The colourful simplicial depth conjecture

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Given d+1 sets of points, or colours, S₁,,Sd₊₁ in Rᵈ, a colourful simplex is a set Tᵢ₌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, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597--604 (2006)) 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.