We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34 (3):497-506, 2005].2000 Mathematics Subject Classification. 52C10 Erdős problems and related topics of discrete geometry, 05D10 Ramsey theory.Key words and phrases. Erdős-Szekeres Theorem, happy end problem, big line or big clique conjecture, empty quadrilateral, empty pentagon, empty hexagon.
Abstract. For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k "walls", represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.
We introduce the study of forcing sets in mathematical origami. The origami material folds flat along straight line segments called creases, each of which is assigned a folding direction of mountain or valley. A subset F of creases is forcing if the global folding mountain/valley assignment can be deduced from its restriction to F . In this paper we focus on one particular class of foldable patterns called Miura-ori, which divide the plane into congruent parallelograms using horizontal lines and zig-zag vertical lines. We develop efficient algorithms for constructing a minimum forcing set of a Miura-ori map, and for deciding whether a given set of creases is forcing or not. We also provide tight bounds on the size of a forcing set, establishing that the standard mountain-valley assignment for the Miura-ori is the one that requires the most creases in its forcing sets. Additionally, given a partial mountain/valley assignment to a subset of creases of a Miura-ori map, we determine whether the assignment domain can be extended to a locally flat-foldable pattern on all the creases. At the heart of our results is a novel correspondence between flat-foldable Miura-ori maps and 3-colorings of grid graphs.
Abstract. The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales it claims that the complement of a chord that uses half the pitches of a scale is homometric to-i.e., has the same interval structure as-the original chord. In terms of onsets it claims that the complement of a rhythm with the same number of beats as rests is homometric to the original rhythm. We generalize the theorem in two directions: from points on a discrete circle (the mathematical model encompassing both scales and rhythms) to a continuous domain, and simultaneously from the discrete presence or absence of a pitch/onset to a continuous strength or weight of that pitch/onset. Athough this is a significant generalization of the Hexachordal Theorem, having all discrete versions as corollaries, our proof is arguably simpler than some that have appeared in the literature.We also establish the natural analog of what is sometimes known as Patterson's second theorem: if two equal-weight rhythms are homometric, so are their complements.
Abstract. This paper studies problems related to visibility among points in the plane. A point x blocks two points v and w if x is in the interior of the line segment vw. A set of points P is k-blocked if each point in P is assigned one of k colours, such that distinct points v, w ∈ P are assigned the same colour if and only if some other point in P blocks v and w. The focus of this paper is the conjecture that each k-blocked set has bounded size (as a function of k). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets {n1, n2, n3, n4} such that some 4-blocked set has exactly ni points in the i-th colour class. Amongst other results, for infinitely many values of k, we construct k-blocked sets with k 1.79... points.2000 Mathematics Subject Classification. 52C10 Erdős problems and related topics of discrete geometry, 05D10 Ramsey theory. This is the full version of an extended abstract to appear in the 26th European Workshop on Computational Geometry (EuroCG '10).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.