Applications of graph containers in the Boolean lattice

Andrew Treglown, Jozsef Balogh, Adam Zsolt Wagner

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
173 Downloads (Pure)

Abstract

We apply the graph container method to prove a number of counting results for the Boolean lattice inline image. In particular, we:

Give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes in inline image, 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 inline image and give a two-coloured analogue;
Give an asymptotic formula for the number of (p, q)-tilted Sperner families in inline image;
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
Original languageEnglish
Pages (from-to)845-872
Number of pages28
JournalRandom Structures and Algorithms
Volume49
Issue number4
Early online date22 Jul 2016
DOIs
Publication statusPublished - Dec 2016

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