Some heterochromatic theorems for matroids

Abstract The anti-Ramsey number of Erdos, Simonovits and Sos from 1973 has become a classic invariant in Graph Theory. To extend this invariant to Matroid Theory, we use the heterochromatic number h c ( H ) 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 ( M ) , whose hyperedges are the circuits of M , or B ( M ) , whose hyperedges are the bases of M . We determine h c ( C ( M ) ) for general matroids and characterise the matroids with the property that h c ( 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.