Fair division with multiple pieces

Given a set of $p$ players we consider problems concerning envy-free allocation of collections of $k$ pieces from a given set of goods or chores. We show that if $p\le n$ and each player can choose $k$ pieces out of $n$ pieces of a cake, then there exist a division of the cake and an allocation of the pieces where at least $\frac{p}{2(k^2-k+1)}$ players get their desired $k$ pieces each. We further show that if $p\le k(n-1)+1$ and each player can choose $k$ pieces, one from each of $k$ cakes that are divided into $n$ pieces each, then there exist a division of the cakes and allocation of the pieces where at least $\frac{p}{2k(k-1)}$ players get their desired $k$ pieces. Finally we prove that if $p\ge k(n-1)+1$ and each player can choose one shift in each of $k$ days that are partitioned into $n$ shifts each, then, given that the salaries of the players are fixed, there exist $n(1+\ln k)$ players covering all the shifts, and moreover, if $k=2$ then $n$ players suffice. Our proofs combine topological methods and theorems of F\"uredi, Lov\'asz and Gallai from hypergraph theory.