Applications of graph containers in the Boolean lattice

TL;DR

This paper employs the graph container method to derive new counting results and bounds for structures within the Boolean lattice, including error-correcting codes, antichains, and Sperner families, along with disproving some conjectures.

Contribution

It introduces novel applications of the graph container method to Boolean lattice problems, providing new bounds, alternative proofs, and counterexamples to existing conjectures.

Findings

Estimated the number of $t$ error correcting codes in $ ext{Boolean lattice}$

Provided an alternative proof of Kleitman's theorem on antichains

Disproved two conjectures on maximal independent sets and antichains

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.