SHAGNIK DAS, WENYING GAN and BENNY SUDAKOV
نویسندگان
چکیده
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, F 1 ⊂ F 2. Erd˝ os extended this theorem to determine the largest family without a k-chain, F 1 ⊂ F 2 ⊂ · · · ⊂ F k. Erd˝ os 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.
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The minimum number of disjoint pairs in set systems and related problems
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