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The heterochromatic number hc(H) of a non-empty hypergraph H is the smallestinteger k such that for every colouring of the vertices of H with exactly kcolours, there is a hyperedge of H all of whose vertices have differentcolours. We denote by nu(H) the number of vertices of H and by tau(H) the sizeof the smallest set containing at least two vertices of each hyperedge of H.For a complete geometric graph G with n > 2 vertices let H = H(G) be thehypergraph whose vertices are the edges of G and whose hyperedges are the edgesets of plane spanning trees of G. We prove that if G has at most one interiorvertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) -tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set givenby the ground set and the bases of a matroid, respectively.

On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids