Related papers: Erdos-Szekeres tableaux
Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$,…
The Erd\H{o}s similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences.…
We study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences. First, we provide sufficient dependence conditions under which these order statistics converges almost surely to…
Given a set of $n$ points in $R^2$, the Szemer\'edi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the…
Erd\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is…
This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all…
An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment…
The Erd\H{o}s-Ginzburg-Ziv theorem states that every sequence of 2n - 1 integers contains a subsequence of length n whose sum is divisible by n. Choi, Kang, and Lim gave a simple deterministic O(n log n) algorithm for finding such a…
A new algorithm named EXPected Similarity Estimation (EXPoSE) was recently proposed to solve the problem of large-scale anomaly detection. It is a non-parametric and distribution free kernel method based on the Hilbert space embedding of…
In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which…
Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n,k) is the graph with vertex set $\binom{[n]}{k}$, such that two vertices are adjacent if they are disjoint. We…
The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose…
Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…
The Erd\H{o}s--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences…
We consider encoding problems for range queries on arrays. In these problems the goal is to store a structure capable of recovering the answer to all queries that occupies the information theoretic minimum space possible, to within lower…
One of the most classical results in Ramsey theory is the theorem of Erd\H{o}s and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following…
In this paper we study pattern-replacement equivalence relations on the set $S_n$ of permutations of length $n$. Each equivalence relation is determined by a set of patterns, and equivalent permutations are connected by pattern-replacements…
The NASA Star and Exoplanet Database (NStED) is a general purpose stellar archive with the aim of providing support for NASA's planet finding and characterization goals, stellar astrophysics, and the planning of NASA and other space…
Let A be a set of integers dense in a finite interval. We establish upper and lower bounds for the longest regularly-spaced and convex subsets of A and of A-A.
The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…