Related papers: Improved Bounds on the Stretch Factor of $Y_4$

We improve the upper bound on the Ramsey number $R(5,5)$ from $R(5,5) \le 49$ to $R(5,5) \le 48$. We also complete the catalogue of extremal graphs for $R(4,5)$.

Motivated by applications of Gabriel graphs and Yao graphs in wireless ad-hoc networks, we show that the maximal degree of a random Gabriel graph or Yao graph defined on $n$ points drawn uniformly at random from a unit square grows as…

We revisit the classical Unit Distance Problem posed by Erd\H{o}s in 1946. While the upper bound of $O(n^{4/3})$ established by Spencer, Szemer'edi, and Trotter (1984) is tight for systems of pseudo-circles, it fails to account for the…

We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact…

Bennett, Carbery and Tao considered the $k$-linear restriction estimate in $\mathbb{R}^{n+1}$ and established the near optimal $L^\frac2{k-1}$ estimate under transversality assumptions only. We have shown that the trilinear restriction…

For the inelastic diffraction, we obtain an upper bound valid in the whole range of the elastic scattering amplitude variation allowed by unitarity. We discuss the energy dependence of the inelastic diffractive cross-section on the base of…

In the paper "The best m-term approximation and greedy algorithms" (V. N. Temlyakov), an error bound for a near best m-term approximation of a function g in L^p([0,1]^d) is provided, using a basis L^p-equivalent to the Haar system, where p…

In this paper we present an extension of known semidefinite and linear programming upper bounds for spherical codes and consider a version of this bound for distance graphs. We apply the main result for the distance distribution of a…

We prove that the finite field Fourier extension operator for the paraboloid is bounded from $L^2\to L^r$ for $r\geq \frac{2d+4}{d}$ in even dimensions $d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal range $r>…

Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$. The problem of bounding $\lambda_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov,…

The optimal exponentials of the thickness in the geometry rigidity inequality of shells represent the geometry rigidity of the shells. We obtain that the lower bounds of the optimal exponentials are $4/3,$ $3/2,$ and $1,$ for the hyperbolic…

A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…

For a finite simple graph $G$ we give an upper bound for the regularity of the powers of the edge ideal $I(G)$.

Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several…

Let $X:=X_n\cup\{(0,0),(1,0)\}$, where $X_n$ is a planar Poisson point process of intensity $n$. We provide a first non-trivial lower bound for the distance between the expected length of the shortest path between $(0,0)$ and $(1,0)$ in the…

Consider the Cayley graph of $S_n$ generated by a random pair of elements $x,y$. Conjecturally, the girth of this graph is $\Omega(n \log n)$ with probability tending to $1$ as $n\to\infty$. We show that it is at least $\Omega(n^{1/3})$.

In this paper we give an improved upper bound, as compared to the one given in [3] for the number of extreme points of the convex set of all G-invariant probability measures on X*Y with given marginals of full support.

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices…

We consider the thickness $\theta (G))$ and outerthickness $\theta _o(G)$ of a graph G in terms of its orientable and nonorientable genus. Dean and Hutchinson provided upper bounds for thickness of graphs in terms of their orientable genus.…

Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u…