  We study a new but simple model for online fair division in which indivisible
items arrive one-by-one and agents have monotone utilities over bundles of the
items. We consider axiomatic properties of mechanisms for this model such as
strategy-proofness, envy-freeness, and Pareto efficiency. We prove a number of
impossibility results that justify why we consider relaxations of the
properties, as well as why we consider restricted preference domains on which
good axiomatic properties can be achieved. We propose two mechanisms that have
good axiomatic fairness properties on restricted but common preference domains.
