  Matroid is a generalization of many fundamental objects in combinatorial
mathematics , and matroid intersection problem is a classical subject in
combinatorial optimization . However , only the intersection of two matroids
are well understood . The solution of the intersection problem of more than
three matroids is proved to be \textbf{NP-hard} . We will give a lower bound
estimate on the maximal cardinality of the common independent sets in matroid
intersections . We will also study some properties of the intersection of more
than two matroids and deduce some analogous results for Edmonds' Min-max
theorems for matroids intersection .
