We consider the problem of fairly allocating indivisible goods to couples, where each couple consists of two agents with distinct additive valuations. We show that there exist instances of allocating indivisible items to $n$ couples for which envy-freeness up to $Ω(\sqrt{n})$ items cannot be guaranteed. This closes the gap by matching the upper bound of Manurangsi and Suksompong, which applies to arbitrary instances with $n$ agents in total. This result is somewhat surprising, as that upper bound was conjectured not to be tight for instances consisting only of small groups.

翻译：我们研究将不可分割物品公平分配给情侣的问题，其中每对情侣由两个具有不同加法估值函数的智能体组成。我们证明，在将不可分割物品分配给$n$对情侣的实例中，存在无法保证消除$Ω(\sqrt{n})$件物品嫉妒的分配方案。这一结果与Manurangsi和Suksompong提出的上界相匹配，从而填补了理论空白——该上界适用于总共有$n$个智能体的任意实例。这一结论略显意外，因为此前学界推测该上界对于仅包含小规模群体的实例并非紧确。