The Boolean lattice is the family of all subsets of ordered by inclusion, and a chain is a family of pairwise comparable elements of . Let , which is the average size of a chain in a minimal chain decomposition of . We prove that can be partitioned into chains such that all but at most proportion of the chains have size . This asymptotically proves a conjecture of Füredi from 1985. Our proof is based on probabilistic arguments. To analyze our random partition we develop a weighted variant of the graph container method. Using this result, we also answer a Kalai-type question raised recently by Das, Lamaison, and Tran. What is the minimum number of forbidden comparable pairs forcing that the largest subfamily of not containing any of them has size at most ? We show that the answer is . Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory.

中文翻译：

---

均匀链分解和应用

布尔格是按包含排序的所有子集的族，而链是 的成对可比元素的族。Let ，它是 的最小链分解中链的平均大小。我们证明可以划分成链，使得除了最多比例的链之外，所有链都具有大小。这渐近地证明了 Füredi 从 1985 年开始的猜想。我们的证明是基于概率论的。为了分析我们的随机分区，我们开发了图容器方法的加权变体。利用这个结果，我们还回答了 Das、Lamaison 和 Tran 最近提出的一个 Kalai 类型的问题。强迫最大亚科的禁止可比对的最小数量是多少不包含它们中的任何一个最多有大小吗？我们证明答案是。最后，我们讨论了如何使用这些均匀链分解来优化和简化极值集理论中的各种结果。