Ron Aharoni, Colorful KKM and multiple cakes division

In the “cake partition” problem n players have each a list of preferred parts for any partition of the [0,1] interval (“cake”) into n sub-intervals. Woodall, Stromquist and Gale proved independently that under mild conditions on the list of preferences (like continuity) there is always a partition and assignment of parts to the players, in which every player gets a piece belonging to her list of preferred parts. In fact, Gale proved a colorful version of the famous KKM theorem, not realizing that this is the same problem, but on the other hand, proved the problem its proper setting. I will discuss the case of partitioning more than one cake – how many players can you make happy, when there is a general number of cakes, and general number of players.

Joint work with Eli Berger, Joseph Briggs, Erel Segal-Halevi and Shira Zerbib.