Applications of graph containers in the Boolean lattice

Abstract

We apply the graph container method to prove a number of counting results for the Boolean lattice $\mathcal P(n)$. In particular, we: (i) Give a partial answer to a question of Sapozhenko estimating the number of $t$ error correcting codes in $\mathcal P(n)$, 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 $\mathcal P(n)$ and give a two-coloured analogue; (iii) Give an asymptotic formula for the number of $(p,q)$-tilted Sperner families in $\mathcal P(n)$; (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.