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A Doubly Exponentially Crumbled Cake

We consider the following cake cutting game: Alice chooses a set P of n points in the square (cake) [0,1]^2, where (0,0) is in P; Bob cuts out n axis-parallel rectangles with disjoint interiors, each of them having a point of P as the lower left corner; Alice keeps the rest. It has been conjectured that Bob can always secure at least half of the cake. This remains unsettled, and it is not even known whether Bob can get any positive fraction independent of n. We prove that if Alice can force Bob's share to tend to zero, then she must use very many points; namely, to prevent Bob from gaining more than 1/r of the cake, she needs at least 2^{2^{Ω(r)}} points.