Supersaturation in the Boolean lattice

Andrew P. Dove 1 Jerrold R. Griggs 1 Ross J. Kang 2 Jean-Sébastien Sereni 3
3 ORPAILLEUR - Knowledge representation, reasonning
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : We prove a "supersaturation-type'' extension of both Sperner's Theorem (1928) and its generalization by Erdos (1945) to k-chains. Our result implies that a largest family whose size is x more than the size of a largest k-chain free family and that contains the minimum number of k-chains is the family formed by taking the middle (k-1) rows of the Boolean lattice and x elements from the k-th middle row. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951).
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Andrew P. Dove, Jerrold R. Griggs, Ross J. Kang, Jean-Sébastien Sereni. Supersaturation in the Boolean lattice. Integers : Electronic Journal of Combinatorial Number Theory, State University of West Georgia, Charles University, and DIMATIA, 2014, The Dick de Bruijn Memorial Volume, 14A, pp.A4. ⟨hal-00802000⟩

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