Related papers: Erdos-Szekeres tableaux
We consider point sets in the real projective plane $\mathbb{R}P^2$ and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erd\H{o}s--Szekeres-type problems. We provide…
We describe an algorithm for compressing a partially ordered set, or \emph{poset}, so that it occupies space matching the information theory lower bound (to within lower order terms), in the worst case. Using this algorithm, we design a…
Recent advances in stochastic differential equations (SDEs) have enabled robust modeling of real-world dynamical processes across diverse domains, such as finance, health, and systems biology. However, parameter estimation for SDEs…
An elliptic divisibility sequence (EDS) is a sequence of integers W_0,W_1,W_2,... generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of W_n for unbounded…
In this paper, we provide new approximation algorithms for dynamic variations of the longest increasing subsequence (\textsf{LIS}) problem, and the complementary distance to monotonicity (\textsf{DTM}) problem. In this setting, operations…
A toric arrangement is an arrangement of subtori of codimension one in a real or complex torus. The poset of layers is the set of connected components of non-empty intersections of these subtori, partially ordered by reverse inclusion. In…
We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this…
Suppose a graph $G$ is stochastically created by uniformly sampling vertices along a line segment and connecting each pair of vertices with a probability that is a known decreasing function of their distance. We ask if it is possible to…
Prediction based on Irregularly Sampled Time Series (ISTS) is of wide concern in the real-world applications. For more accurate prediction, the methods had better grasp more data characteristics. Different from ordinary time series, ISTS is…
The Erd\H{o}s-Szekeres conjecture states that any set of more than $2^{n-2}$ points in the plane with no three on a line contains the vertices of a convex $n$-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that any…
This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szab\'{o} Theorem…
We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$ \sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right| $$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the…
Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for pattern-restricted permutations in…
In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the…
We extend the famous Erd\H{o}s-Szekeres theorem to $k$-flats in ${\mathbb{R}^d}$
In this paper, we study gradient-based classical extremum seeking (ES) for uncertain n-dimensional (nD) static quadratic maps in the presence of known large constant distinct input delays and large output constant delay with a small…
In 2001, K\'arolyi, Pach and T\'oth introduced a family of point sets to solve an Erd\H{o}s-Szekeres type problem; which have been used to solve several other Ed\H{o}s-Szekeres type problems. In this paper we refer to these sets as nested…
Let OT_d(n) be the smallest integer N such that every N-element point sequence in R^d in general position contains an order-type homogeneous subset of size n, where a set is order-type homogeneous if all (d+1)-tuples from this set have the…
This paper investigates several classical and novel variations of the Erd\H{o}s--Szekeres problem, including multicolored point sets, convex hexagons with a given number of interior points, and polygons with constraints on edge colors. We…
In this paper, we provide an overview of Ehrhart polynomials associated with order polytopes of finite posets, a concept first introduced by Stanley. We focus on their combinatorial interpretations for many sequences listed on the OEIS. We…