On the Existence of Fair Allocations for Goods and Chores under Dissimilar Preferences

We study the fundamental problem of fairly allocating a multiset $\mathcal{M}$ of $t$ types of indivisible items among $d$ groups of agents, where all agents within a group have identical additive valuations. Gorantla et al. [GMV23] showed that for every such instance, there exists a finite number $\mu$ such that, if each item type appears at least $\mu$ times, an envy-free allocation exists. Their proof is non-constructive and only provides explicit upper bounds on $\mu$ for the cases of two groups ($d=2$) or two item types ($t=2$). In this work, we resolve one of the main open questions posed by Gorantla et al. [GMV23] by deriving explicit upper bounds on $\mu$ that hold for arbitrary numbers of groups and item types. We introduce a significantly simpler, yet powerful technique that not only yields constructive guarantees for indivisible goods but also extends naturally to chores and continuous domains, leading to new results in related fair division settings such as cake cutting.