Motivated by the celebrated Beck‐Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an discrepancy bound when n ≤ m and an O(1) bound when n ≫ m^t. In this paper, we give a tight bound for the entire range of n and m, under a mild assumption that . The result is based on two steps. First, applying the partial coloring method to the case when and using the properties of the random set system we show that the overall discrepancy incurred is at most . Second, we reduce the general case to that of using LP duality and a careful counting argument.

中文翻译：

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关于随机低度集系统的差异

受著名的贝克·菲亚拉猜想的启发，我们考虑存在n个元素和m个集合，每个元素位于t个随机选择的集合中的随机设置。在该设置中，以斯拉和维特呈结合时差异Ñ  ≤ 米和ø结合时（1）ñ  » 米^吨。在本文中，我们在n和m的整个范围内给出了一个严格的边界，即。结果基于两个步骤。首先，将部分着色方法应用于并利用随机集系统的性质，我们表明所产生的总体差异最多。其次，我们将一般情况简化为使用LP对偶和仔细计数的论点。