Abstract Single minded agents have strict preferences, in which a bundle is acceptable only if it meets a certain demand. Such preferences arise naturally in scenarios such as allocating computational resources among users, where the goal is to fairly serve as many requests as possible. In this paper we study the fair division problem for such agents, which is complex due to discontinuity and complementarities of preferences. Our solution concept—the competitive allocation from equal incomes (CAEI)—is inspired from market equilibria and implements fair outcomes through a pricing mechanism. We study existence and computation of CAEI for multiple divisible goods, discrete goods, and cake cutting. Our solution is useful more generally, when the players have a target set of goods, and very small positive values for any bundle other than their target set.