Some heterochromatic theorems for matroids

The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory. To extend this invariant to Matroid Theory, we use the heterochromatic number $hc\left(H\right)$ of a non-empty hypergraph $H$. The heterochromatic number of $H$ is the smallest integer $k$ such that for every colouring of the vertices of $H$ with exactly $k$ colours, there is a totally multicoloured hyperedge of $H$.Given a matroid $M$, there are several hypergraphs over the ground set of $M$ we can consider, for example, $C\left(M\right)$, whose hyperedges are the circuits of $M$, or $B\left(M\right)$, whose hyperedges are the bases of $M$. We determine $hc\left(C\left(M\right)\right)$ for general matroids and characterise the matroids with the property that $hc\left(B\left(M\right)\right)$ 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.