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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 twocoloured analogue;
Give an asymptotic formula for the number of (p, q)tilted Sperner families in inline image;
Prove a random version of Katona's tintersection 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
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 twocoloured analogue;
Give an asymptotic formula for the number of (p, q)tilted Sperner families in inline image;
Prove a random version of Katona's tintersection 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 language  English 

Pages (fromto)  845872 
Number of pages  28 
Journal  Random Structures and Algorithms 
Volume  49 
Issue number  4 
Early online date  22 Jul 2016 
DOIs  
Publication status  Published  Dec 2016 
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Dive into the research topics of 'Applications of graph containers in the Boolean lattice'. Together they form a unique fingerprint.Projects
 1 Finished

EPSRC Fellowship: Dr Andrew Treglown  Independence in groups, graphs and the integers
Engineering & Physical Science Research Council
1/06/15 → 31/05/18
Project: Research Councils