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Several fundamental results of combinatorial optimization, such as [[Edmonds' matching polytope theorem]], or the [[Edmonds-Giles theorem]], deal with the [[integer polyhedron|integrality of polyhedra]]. These results have been generalized in several directions, and today we have a vast knowledge of the integrality properties of polyhedra defined by various types of constraints. However, there are some really elegant conjectures that are unresolved, and which suggest that there may still be deep theoretical insights to be made.