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A collection of objects, some of which are good and some are bad, is to be divided fairly among agents with different tastes, modeled by additive utility functions. If the objects cannot be shared, so that each of them must be entirely allocated to a single agent, then a fair division may not exist. What is the smallest number of objects that must be shared between two or more agents in order to attain a fair and efficient division? In this paper, fairness is understood as proportionality orarXiv:1908.01669v3 fatcat:xyul32nzfjcq3blnn7ag5n6uqu