Related papers: Explicit universal minimal constants for polynomia…

In many interacting particle systems, relaxation to equilibrium is thought to occur via the growth of 'droplets', and it is a question of fundamental importance to determine the critical length at which such droplets appear. In this paper…

Let $\Gamma$ be a finitely generated subgroup of the multiplicative group $\G_m^2(\bar{Q})$. Let $p(X,Y),q(X,Y)\in\bat{Q}$ be two coprime polynomials not both vanishing at $(0,0)$; let $\epsilon>0$. We prove that, for all $(u,v)\in\Gamma$…

We show that an arbitrary nilprogression can be approximated by a proper coset nilprogression in upper-triangular form. This can be thought of as a nilpotent version of the Freiman-Bilu result that a generalised arithmetic progression can…

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the…

Let $\gamma(t)=(P_1(t),\ldots,P_n(t))$ where $P_i$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+\gamma(t)\}$ in sets of positive density $\epsilon$ in…

We provide optimal upper bounds on the growth of iterated sumsets $hA=A+\dots+A$ for finite subsets $A$ of abelian semigroups. More precisely, we show that the new upper bounds recently derived from Macaulay's theorem in commutative algebra…

For any positive integers $n\ge d+1\ge 3$, what is the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound…

In $r$-neighbor bootstrap percolation on the vertex set of a graph $G$, a set $A$ of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least $r$ previously infected neighbors. When the…

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in…

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a…

We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean…

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$,…

Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and…

We investigate the maximal degree in a Poisson-Delaunay graph in $\mathbf{R}^d$, $d\geq 2$, over all nodes in the window $\mathbf{W}_\rho:= \rho^{1/d}[0,1]^d$ as $\rho$ goes to infinity. The exact order of this maximum is provided in any…

Given a group G, the model $\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. In this article we show that for any…

We prove two results concerning percolation on general graphs. - We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies $p_c<1$, then the number of minimal cutsets of size…

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and…

For $d \geq 2$ and $n \in \mathbb{N}$ even, let $p_n = p_n(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^d$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n \in…

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

Dallard, Milani\v{c}, and \v{S}torgel conjectured that for a hereditary graph class $\mathcal{G}$, if there is some function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G\in \mathcal{G}$ with clique number $\omega(G)$ has treewidth…