Uniform chain decompositions and applications
نویسندگان
چکیده
The Boolean lattice is the family of all subsets ordered by inclusion, and a chain pairwise comparable elements . Let , which average size in minimal decomposition We prove that can be partitioned into chains such but at most proportion have This asymptotically proves conjecture Füredi from 1985. Our proof based on probabilistic arguments. To analyze our random partition we develop weighted variant graph container method. Using this result, also answer Kalai-type question raised recently Das, Lamaison, Tran. What minimum number forbidden pairs forcing largest subfamily not containing any them has ? show Finally, discuss how these uniform decompositions used to optimize simplify various results extremal set theory.
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ژورنال
عنوان ژورنال: Random Structures and Algorithms
سال: 2021
ISSN: ['1042-9832', '1098-2418']
DOI: https://doi.org/10.1002/rsa.21034