Related papers: Upper tail bounds for Stars

In this paper we consider a number of natural decision problems involving k-regular sequences. Specifically, they arise from - lower and upper bounds on growth rate; in particular boundedness, - images, - regularity (recognizability by a…

Harary's conjecture $r(C_3,G)\leq 2q+1$ for every isolated-free graph G with $q$ edges was proved independently by Sidorenko and Goddard and Klietman. In this paper instead of $C_3$ we consider $K_{2,k}$ and seek a sharp upper bound for…

A fan $F_n$ is a graph consisting of $n$ triangles, all having precisely one common vertex. Currently, the best known bounds for the Ramsey number $R(F_n)$ are $9n/2-5 \leq R(F_n) \leq 11n/2+6$, obtained by Chen, Yu and Zhao. We improve the…

This paper addresses the advancement of probability tail bound analysis, a crucial statistical tool for assessing the probability of large deviations of random variables from their expected values. Traditional tail bounds, such as Markov's,…

Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of…

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing…

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 $c_k > 0$,…

We consider the supercritical bond percolation on $\mathbb Z^d$ and study the graph distance on the percolation graph called the chemical distance. It is well-known that there exists a deterministic constant $\mu(x)$ such that the chemical…

We provide the first tight bounds on the lower tail probability of the one point distribution of the KPZ equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between…

We derive exponential bounds for tail of distribution for natural, i.e. under ordinary logarithm, normalized sums of arrays of random variables, not necessarily independent.

The $n$-star $S_n$ is the $n$-vertex triple system with ${n-1 \choose 2}$ edges all of which contain a fixed vertex, and $K_4^-$ is the unique triple system with four vertices and three edges. We prove that the Ramsey number $r(K_4^-, S_n)$…

Let $\Psi_1,\Psi_2,...$ be a sequence of i.i.d. random Lipschitz functions on a complete separable metric space with unbounded metric $d$ and forward iterations $X_n$. Suppose that $X_n$ has a stationary distribution. We study the…

Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its…

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this…

Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$…

Let $N$ be the number of copies of a small subgraph $H$ in an Erd\H{o}s-R\'enyi graph $G \sim \mathcal{G}(n, p_n)$ where $p_n \to 0$ is chosen so that $\mathbb{E} N = c$, a constant. Results of Bollob\'as show that for regular graphs $H$,…

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs…

We study the maximum of the random assignment process on rectangular matrices. We derive first-order asymptotics for the expected maximum, prove a law of large numbers under mild tail assumptions, and obtain exponential upper bounds for the…

Given a graph $F$, we define $\operatorname{ex}(G_{n,p},F)$ to be the maximum number of edges in an $F$-free subgraph of the random graph $G_{n,p}$. Very little is known about $\operatorname{ex}(G_{n,p},F)$ when $F$ is bipartite, with…

We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a result of Krizek, Luca and Somer and give the corrected, slightly weaker upper bound.…