Sperner定理与一个Erdos-Katona-Kleitman问题
Sperner's Theorem and a Problem of Erdos-Katona-Kleitman
摘要：
极值集合论的一个中心结果是1928年著名的Sperner定理，它给出了不包含2-链的[n]子集的最大族的大小。Erdos将这个定理推广到确定没有k链的最大族。Erdos和Katona，其次是Kleitman，问有多少链必须出现在家庭的大小大于相应的极值界限。在1966年，Kleitman解决了这个问题的2-链，表明这样的链的数量是最小化，通过采取集尽可能接近中间水平。此外，他还证明了对于所有k-链，极值族都是相同的。本文在这个问题上取得了初步的进展，我们对至多为$k+1$中间能级的族证明了Kleitman猜想.我们还描述了所有的极值配置。
英文摘要：
A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain. Erdos extended this theorem to determine the largest family without a k-chain. Erdos and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds. In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman's conjecture for the families whose size is at most the size of the $k+1$ middle levels. We also characterize all extremal configurations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
