We study the problem of fairly allocating a set of indivisible items to a set of agents with additive valuations. Recently, Feige et al. (WINE'21) proved that a maximin share (MMS) allocation exists for all instances with $n$ agents and no more than $n + 5$ items. Moreover, they proved that an MMS allocation is not guaranteed to exist for instances with $3$ agents and at least $9$ items, or $n \ge 4$ agents and at least $3n + 3$ items. In this work, we shrink the gap between these upper and lower bounds for guaranteed existence of MMS allocations. We prove that for any integer $c > 0$, there exists a number of agents $n_c$ such that an MMS allocation exists for any instance with $n \ge n_c$ agents and at most $n + c$ items, where $n_c \le \lfloor 0.6597^c \cdot c!\rfloor$ for allocation of goods and $n_c \le \lfloor 0.7838^c \cdot c!\rfloor$ for chores. Furthermore, we show that for $n \neq 3$ agents, all instances with $n + 6$ goods have an MMS allocation.

翻译：我们研究在加性估值下将一组不可分割物品公平分配给一组智能体的问题。近期，Feige等人（WINE'21）证明，对于任意包含$n$个智能体且物品数量不超过$n + 5$的实例，最大最小份额（MMS）分配必然存在。此外，他们证明当实例包含3个智能体且至少9件物品，或$n \ge 4$个智能体且至少$3n + 3$件物品时，MMS分配无法保证存在。本文缩小了MMS分配存在性保证的上界与下界之间的差距。我们证明：对于任意整数$c > 0$，存在智能体数量$n_c$，使得当实例包含$n \ge n_c$个智能体且物品数量不超过$n + c$时，MMS分配必然存在；其中对于物品分配，$n_c \le \lfloor 0.6597^c \cdot c!\rfloor$，对于家务分配，$n_c \le \lfloor 0.7838^c \cdot c!\rfloor$。进一步，我们证明当$n \neq 3$时，所有包含$n + 6$件物品的实例均存在MMS分配。