  Given a combinatorial optimization problem $\Pi$ and an increasing finite
sequence $c$ of natural numbers, we obtain a cardinality constrained version
$\Pi_c$ of $\Pi$ by permitting only those feasible solutions of $\Pi$ whose
cardinalities are members of $c$. We are interested in polyhedra associated
with those problems, in particular in inequalities that cut off solutions of
forbidden cardinality. Maurras (1977) and Camion and Maurras (1982) introduced
a family of inequalities, that we call forbidden set inequalities, which can be
used to cut off those solutions. However, these inequalities are in general not
facet defining for the polyhedron associated with $\Pi_c$. Kaibel and Stephan
(2007) showed how one can combine integer characterizations for cycle and path
polytopes and a modified form of forbidden set inequalities to give facet
defining integer representations for the cardinality restricted versions of
these polytopes. Motivated by this work, we apply the same approach on the
matroid polytope. It is well known that the so-called rank inequalities
together with the nonnegativity constraints provide a complete linear
description of the matroid polytope (see Edmonds (1971). By essentially adding
the forbidden set inequalities in an appropriate form, we obtain a complete
linear description of the cardinality constrained matroid polytope which is the
convex hull of the incidence vectors of those independent sets that have a
feasible cardinality. Moreover, we show how the separation problem for the
forbidden set inequalities can be reduced to that for the rank inequalities. We
also give necessary and sufficient conditions for a forbidden set inequality to
be facet defining.
