Anti-Ramsey number of disjoint rainbow bases in all matroids

Consider a matroid M = (E, I) with its elements of the ground set E colored. A rainbow basis is a maximum independent set in which each element receives a different color. The rank of a subset S of E, denoted by rM (S), is the maximum size of an independent set in S. A flat F is a maximal set in M with a fixed rank. The anti-Ramsey number of t pairwise disjoint rainbow bases in M , denoted by ar(M, t), is defined as the maximum number of colors m such that there exists an m coloring of the ground set E of M which contains no t pairwise disjoint rainbow bases. We determine ar(M, t) for all matroids of rank at least 2: ar(M, t) = |E| if there exists a flat F0 with |E| − |F0| < t(rM (E) − rM (F0)); and ar(M, t) = maxF : rM (F )≤rM (E)−2{|F | + t(rM (E) − rM (F ) − 1)} otherwise. This generalizes Lu-Meier-Wang’s previous result on the anti-Ramsey number of edge-disjoint rainbow spanning trees in any multigraph G.