Envy-free two-player m-cake and three-player two-cake divisions

Cloutier, Nyman, and Su (2005) initiated the study of envy-free cake-cutting problems involving several cakes. They showed that when there are two players and two or three cakes it is possible to find envy-free cake-divisions requiring few cuts, under natural assumptions. We prove that such a result also exists when there are two players and any number of cakes and when there are three players and two cakes. The proof relies on the fractional matching number in m-partite hypergraphs.