PROBLEM 5. Let M = (E, F) be a matroid with rank function r. The span of S C E is span (S) = {j EE : r(SUj) = r(S) }. Let Mi = (E, Fi), i= 1, ..., k be k copies of the same matroid M. Let J be a maximum cardinality partitionable set. Show that there are disjoint independent sets F1, F2, . . . FK such that E\\ J C span (F1) =... = span(Fk). PROBLEM 6. Use the Edmonds polytope and LP duality to derive the Tutte-Berge formula