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A probabilistic variant of Sperner ’s theorem and of maximal r-cover free families
Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-10-01 , DOI: 10.1016/j.disc.2020.112027
Noga Alon , Shoni Gilboa , Shay Gueron

Abstract A family of sets is called r -cover free if no set in the family is contained in the union of r (or less) other sets in the family. A 1-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner’s classical result determines the maximal cardinality of a 1-cover free family of subsets of an n -element set. Estimating the maximal cardinality of an r -cover free family of subsets of an n -element set for r > 1 was also studied. In this note we are interested in the following probabilistic variant of this problem. Let S 0 , S 1 , … , S r be independent and identically distributed random subsets of an n -element set. Which distribution minimizes the probability that S 0 ⊆ ⋃ i = 1 r S i ? A natural candidate is the uniform distribution on an r -cover-free family of maximal cardinality. We show that for r = 1 such distribution is indeed best possible. In a complete contrast, we also show that this is far from being true for every r > 1 and n large enough.

中文翻译:

Sperner 定理和最大 r 覆盖自由族的概率变体

摘要 如果一个集合族中没有包含在该族中 r 个(或更少)个其他集合的并集中,则该集合族被称为 r -cover free。一个 1-cover free family 只是一个关于集合包含的反链。因此,Sperner 的经典结果确定了 n 元素集的 1 覆盖自由子集族的最大基数。还研究了估计 r > 1 的 n 元素集的无 r 覆盖的子集族的最大基数。在本笔记中,我们对该问题的以下概率变体感兴趣。令 S 0 , S 1 , … , S r 是 n 元素集合的独立同分布随机子集。哪个分布使 S 0 ⊆ ⋃ i = 1 r S i 的概率最小?一个自然的候选者是最大基数的无 r 覆盖族的均匀分布。我们表明对于 r = 1 这种分布确实是最好的。相比之下,我们还表明,对于每个 r > 1 和足够大的 n,这远非如此。
更新日期:2020-10-01
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