We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets
in the discrete hypercube {0, 1} d and Galvin's follow‐up work on weighted independent …

By implementing algorithmic versions of Sapozhenko's graph container methods, we give
new algorithms for approximating the number of independent sets in bipartite graphs. Our …

Given a connected finite graph $ G $, an integer-valued function $ f $ on $ V (G) $ is called $
M $-Lipschitz if the value of $ f $ changes by at most $ M $ along the edges of $ G $. In 2013 …

We study the locations of complex zeroes of independence polynomials of bounded-degree
hypergraphs. For graphs, this is a long-studied subject with applications to statistical …

For an odd integer n= 2 d− 1, let B (n, d) be the subgraph of the hypercube Q n induced by
the two largest layers. In this paper, we describe the typical structure of independent sets in …

Dedekind's problem, dating back to 1897, asks for the total number $\psi (n) $ of antichains
contained in the Boolean lattice $ B_n $ on $ n $ elements. We study Dedekind's problem …

We study the typical structure and the number of triangle-free graphs with $ n $ vertices and
$ m $ edges where $ m $ is large enough so that a typical triangle-free graph has a cut …

For an odd integer\(n= 2d-1\), let\({\mathcal {B}} _d\) be the subgraph of the
hypercube\(Q_n\) induced by the two largest layers. In this paper, we describe the typical …

We establish long-range order for discrete nearest-neighbor spin systems on $\mathbb {Z}^
d $ satisfying a certain symmetry assumption, when the dimension $ d $ is higher than an …

Let $ Q_n $ be the $ n $-dimensional Hamming cube and $ N= 2^ n $. We prove that the
number of maximal independent sets in $ Q_n $ is asymptotically\[2n2^{N/4},\] as was …