Maximal independent sets in the middle two layers of the Boolean lattice
József Balogh, Ce Chen, Ramon I. Garcia
TL;DR
This work resolves the precise asymptotics for the number of maximal independent sets in the middle two layers $B(2d-1,d)$ of the Boolean lattice by developing a novel variant of Sapozhenko's container method together with new isoperimetric tools. The authors show $\text{MIS}(B(2d-1,d)) = (1+o(1))(2d-1) \exp\left(\frac{(d-1)^2}{2^{2d-1}}\binom{2d-2}{d-1}\right) \cdot 2^{\binom{2d-2}{d-1}}$, and provide a detailed description of the typical structure: almost all MIS come from large induced matchings aligned in one direction, with a canonical matching $M_k$ capturing the main interaction. The approach blends a refined container method, a new isoperimetric lemma based on shifting, stability results from Hujter–Tuza and Kahn–Park, and careful counting to isolate the dominant configurations. These results illuminate the typical structure of MIS in layered hypercube graphs and open avenues for applying the modified container framework to related counting problems in the hypercube and other graph families.
Abstract
Let $B(2d-1, d)$ be the subgraph of the hypercube $\mathcal{Q}_{2d-1}$ induced by its two largest layers. Duffus, Frankl and Rödl proposed the problem of finding the asymptotics for the logarithm of the number of maximal independent sets in $B(2d-1, d)$. Ilinca and Kahn determined the logarithmic asymptotics and reiterated the question of what their order of magnitude is. We show that the number of maximal independent sets in $B(2d-1,d)$ is \[ \left(1+o(1)\right)(2d-1)\exp\left(\frac{(d-1)^2}{2^{2d-1}}\binom{2d-2}{d-1}\right)\cdot 2^{\binom{2d-2}{d-1}}, \] and describe their typical structure. The proof uses a new variation of Sapozhenko's Graph Container Lemma, a new isoperimetric lemma, a theorem of Hujter and Tuza on the number of maximal independent sets in triangle-free graphs and a stability version of their result by Kahn and Park, among other tools.