  Consider a matroid $M=(E,\mathcal{I})$ 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
$r_M(S)$, 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 $F_0$ with $|E|-|F_0|<t(r_M(E)-r_M(F_0))$; and $ar(M,t)=\max_{F\colon
r_M(F)\leq r_M(E)-2} \{|F|+t(r_M(E)-r_M(F)-1)\}$ otherwise. This generalizes
Lu-Meier-Wang's previous result on the anti-Ramsey number of edge-disjoint
rainbow spanning trees in any multigraph $G$.
