Related papers: Independent sets of a given size and structure in …

In this expository note we describe a proof due to A. Sapozhenko that the number of independent sets in the discrete $d$-dimensional hypercube $Q_d$ is asymptotically $2 \sqrt{e} 2^{2^{d-1}}$ as $d$ tends to infinity.

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 sets. We combine Sapozhenko's graph container methods with…

Let $Q_{d,p}$ be the random subgraph of the $d$-dimensional hypercube $\{0,1\}^d$, where each edge is retained independently with probability $p$. We study the asymptotic number of independent sets in $Q_{d,p}$ as $d \to \infty$ for a wide…

For an odd integer $n=2d-1$, let $\mathcal{B}(n, d)$ 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 $\mathcal{B}(n, d)$ and give precise…

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…

Let $I$ be an independent set drawn from the discrete $d$-dimensional hypercube $Q_d=\{0,1\}^d$ according to the hard-core distribution with parameter $\lambda>0$ (that is, the distribution in which each independent set $I$ is chosen with…

In this paper we provide an asymptotic expansion for the number of independent sets in a general class of regular, bipartite graphs satisfying some vertex-expansion properties, extending results of Jenssen and Perkins on the hypercube and…

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 conjectured by Ilinca and the first author in connection with a question of…

The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been…

Let $Q_d$ be the $d$-dimensional Hamming cube and $N=|V(Q_d)|=2^d$. An independent set $I$ in $Q_d$ is called balanced if $I$ contains the same number of even and odd vertices. We show that the logarithm of the number of balanced…

It is a well known result due to Korshunov and Sapozhenko that the hypercube in $n$ dimensions has $(1 + o(1)) \cdot 2 \sqrt e \cdot 2^{2^{n-1}}$ independent sets. Jenssen and Keevash investigated in depth Cartesian powers of cycles of…

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski…

We prove matching asymptotic lower and upper bounds on the variances of the intrinsic volumes and the number of $k$-faces of $d$-dimensional random beta-polytopes. Using Stein's methods, we establish central limit theorems for the intrinsic…

Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for $S\subseteq V$…

We revisit the problem of determining the independent domination number in hypercubes for which the known upper bound is still not tight for general dimensions. We present here a constructive method to build an independent dominating set…

We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up…

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and…

Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a $d$-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are…

We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing…