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arXiv · 2505.00132

Maximal independent sets in the middle two layers of the Boolean lattice

József Balogh·Ce Chen·Ramon I. Garcia
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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 R\"odl 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.

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József Balogh, Ce Chen, Ramon I. Garcia. 2025-04-30. Maximal independent sets in the middle two layers of the Boolean lattice. https://arxiv.org/abs/2505.00132

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