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Applications of graph containers in the Boolean lattice
Author(s) -
Balogh József,
Treglown Andrew,
Wagner Adam Zsolt
Publication year - 2016
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20666
Subject(s) - mathematics , lattice (music) , combinatorics , discrete mathematics , graph , intersection (aeronautics) , physics , acoustics , engineering , aerospace engineering
We apply the graph container method to prove a number of counting results for the Boolean lattice P ( n ) . In particular, we: Give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes in P ( n ) , and we also give an upper bound on the number of transportation codes; Provide an alternative proof of Kleitman's theorem on the number of antichains in P ( n ) and give a two‐coloured analogue; Give an asymptotic formula for the number of ( p, q )‐tilted Sperner families in P ( n ) ; Prove a random version of Katona's t ‐intersection theorem. In each case, to apply the container method, we first prove corresponding supersaturation results. We also give a construction which disproves two conjectures of Ilinca and Kahn on maximal independent sets and antichains in the Boolean lattice. A number of open questions are also given. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 845–872, 2016