Some heterochromatic theorems for matroids

Given a matroid M, there are several hypergraphs over the ground set of M we can consider, for example, C(M), whose hyperedges are the circuits of M, or B(M), whose hyperedges are the bases of M. We determine hc(C(M)) for general matroids and characterise the matroids with the property that hc(B(M)) equals the rank of the matroid. We also consider the case when the hyperedges are the Hamiltonian circuits of the matroid. Finally, we extend the known result about the anti-Ramsey number of 3-cycles in complete graphs to the heterochromatic number of 3-circuits in projective geometries over finite fields, and we propose a problem very similar to the famous problem on the anti-Ramsey number of the p-cycles.