Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).

翻译：确定积集判定问题的随机化（或分布）通信复杂度是通信复杂度领域的核心问题，其根源可追溯至Babai、Frankl和Simon在20世纪80年代的基础性工作，最终以1992年Kalyanasundaram-Schnitger与Razborov的著名研究达到高峰。然而，关于乘积分布的紧界问题一直悬而未决，直至近期Bottesch、Gavinsky与Klauck的工作才将其解决。本文重新审视这一经典问题，给出最优界的简洁化证明，并在误差参数依赖关系上实现改进的量化结果。我们的方法基于一个可能具有独立价值的简单组合引理：若从两个分布中独立抽取的两组集合以不可忽略概率不相交，则可提取两个测度较大的子族，这些子族完全交叉不相交（等价于积集判定问题中存在大型单色矩形）。