Set-disjointness problems are one of the most fundamental problems in communication complexity and have been extensively studied in past decades. Given its importance, many lower bound techniques were introduced to prove communication lower bounds of set-disjointness. Combining ideas from information complexity and query-to-communication lifting theorems, we introduce a density increment argument to prove communication lower bounds for set-disjointness: We give a simple proof showing that a large rectangle cannot be $0$-monochromatic for multi-party unique-disjointness. We interpret the direct-sum argument as a density increment process and give an alternative proof of randomized communication lower bounds for multi-party unique-disjointness. Avoiding full simulations in lifting theorems, we simplify and improve communication lower bounds for sparse unique-disjointness. Potential applications to be unified and improved by our density increment argument are also discussed.

翻译：集合不相交问题是通信复杂度中最基础的问题之一，在过去几十年中得到了广泛研究。鉴于其重要性，人们引入了许多下界技术来证明集合不相交问题的通信下界。结合信息复杂度和查询到通信提升定理的思想，我们引入一种密度增量论证来证明集合不相交问题的通信下界：我们给出一个简单证明，表明大矩形不可能是多方唯一不相交问题的 $0$-单色。我们将直接和论证解释为密度增量过程，并给出多方唯一不相交问题随机通信下界的替代证明。通过避免提升定理中的完全模拟，我们简化并改进了稀疏唯一不相交问题的通信下界。最后讨论了用我们的密度增量论证进行统一和改进的潜在应用。