Related papers: The Park-Pham Theorem with Optimal Convergence Rat…

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance…

We prove that for all $k \ge 3$ and any integers $\Delta, n$ with $n \ge 2^\Delta,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\Delta$ such that $r(H)\geq\tw_{k-1}(c_k \Delta) \cdot n$ for some constant $c_k > 0$,…

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability…

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

Correlation analysis is a fundamental problem in statistics. In this paper, we consider the correlation detection problem between a pair of Erdos-Renyi graphs. Specifically, the problem is formulated as a hypothesis testing problem: under…

Let $G$ be a graph on $n$ vertices of maximum degree $\Delta$. We show that, for any $\delta > 0$, the down-up walk on independent sets of size $k \leq (1-\delta)\alpha_c(\Delta)n$ mixes in time $O_{\Delta,\delta}(k\log{n})$, thereby…

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…

We study the graph constructed on a Poisson point process in $d$ dimensions by connecting each point to the $k$ points nearest to it. This graph a.s. has an infinite cluster if $k > k_c(d)$ where $k_c(d)$, known as the critical value,…

For $k\ge 2$ and $r\ge 1$ such that $k+r\ge 4$, we prove that, for any $\alpha>0$, there exists $\epsilon>0$ such that the union of an $n$-vertex $k$-graph with minimum codegree $\left(1-\binom{k+r-2}{k-1}^{-1}+\alpha\right)n$ and a…

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

Let $G=(V, E)$ be a given edge-weighted graph and let its {\em realization} $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e \in E$ independently with probability $p$. In the {\em stochastic matching} problem, the goal…

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be…

We establish a general Berry-Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of…

Fix a graph $H$ and some $p\in (0,1)$, and let $X_H$ be the number of copies of $H$ in a random graph $G(n,p)$. Random variables of this form have been intensively studied since the foundational work of Erd\H{o}s and R\'{e}nyi. There has…

In many branches of engineering, Banach contraction mapping theorem is employed to establish the convergence of certain deterministic algorithms. Randomized versions of these algorithms have been developed that have proved useful in…

Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices…

The problem of constructing an optimal co-adapted coupling for a pair of symmetric random walks on $Z_2^d$ was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such…

We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided…

Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.}…

A maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We…