Applications of graph containers in the Boolean lattice

We apply the graph container method to prove a number of counting results for the Boolean lattice P(n). In particular, we: (i) 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; (ii) Provide an alternative proof of Kleitman’s theorem on the number of antichains in P(n) and give a two-coloured analogue; (iii) Give an asymptotic formula for the number of (p, q)-tilted Sperner families in P(n); (iv) 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.