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 P(n). In particular, we: 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; Provide an alternative proof of Kleitman's theorem on the number of antichains in P(n) and give a two‐coloured analogue; Give an asymptotic formula for the number of (p, q)‐tilted Sperner families in P(n); 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

Supersaturation in posets and applications involving the container method

“…To illustrate the applicability of Lemma 2.2, we prove a weak version of the theorem of Kleitman [23] in the boolean lattice which is often good enough for applications involving the container method (see, e.g., [6,8,11] and Theorem 6.1).…”

“…Also, for n ≥ j, define n ≤j := j r=0 n r . A key idea from [6] is that it can often be advantageous to prove a 'multi-stage' container lemma, where the purpose of the early stages is to 'thin out' the poset to reduce the overall number of containers (see also [8]). We prove the following lemma of this type, which implies Lemma 5.2.…”

“…For instance, many of the proofs in [5,35] applied the classical graph supersaturation theorem of Erdős and Simonovits [20]. For additional applications of the container method, see [4,8,31] and the references therein.…”

Random and deterministic versions of extremal poset problems

“…a 2graph), we may employ an algorithm to generate our containers. This algorithm is discussed in [10] where it is credited to Kleitman and Winston [35; 36]. Its outline is presented below as Algorithm 3.2.1.…”

“…Maker-Breaker games [43], calculating Folkman numbers [46], examining H-free and induced H-free graphs [49], proving sparse random versions of famous theorems such as Erdős-Stone [8], and counting error-correcting codes [10]. Many of these results are enlightening because of the additional tools that are developed to facilitate their proofs.…”