Fair division with multiple pieces

Given a set of players, we consider problems concerning envy-free allocation of collections of pieces from given sets of goods or chores. We show that if and each player prefers pieces in any division of a cake into pieces, then there exists a division of the cake where at least players get their desired pieces each. We further show that if and each player prefers pieces, one piece from each of cakes, in any division of the cakes into pieces each, then there exists a division of the cakes where at least players get their desired pieces each. Finally we prove that if and each player prefers one shift in each of days that are partitioned into shifts each, then, given that players prefer empty shifts if possible (e.g., if salaries are fixed and do not depend on the number of working hours), there exist players covering all the shifts, and moreover, if then players suffice. Our proofs combine topological methods and theorems of Füredi, Lovász and Gallai from hypergraph theory.