Curves in R^d intersecting every hyperplane at most d+1 times

By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing curves in R^d are often called convex curves and they form an important class; a primary example is the moment curve {(t,t^2,...,t^d):t\in[0,1]}. They are also closely related to Chebyshev systems, which is a notion of considerable importance, e.g., in approximation theory. We prove that for every d there is M=M(d) such that every at most d+1 crossing curve in R^d can be subdivided into at most M convex curves. As a consequence, based on the work of Elias, Roldan, Safernova, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in R^d concerning order-type homogeneous sequences of points, investigated in several previous papers.Comment: Corrected proof of Lemma 3.

Track Layouts of Graphs

A \emph{$(k,t)$-track layout} of a graph $G$ consists of a (proper) vertex $t$-colouring of $G$, a total order of each vertex colour class, and a (non-proper) edge $k$-colouring such that between each pair of colour classes no two monochromatic edges cross. This structure has recently arisen in the study of three-dimensional graph drawings. This paper presents the beginnings of a theory of track layouts. First we determine the maximum number of edges in a $(k,t)$-track layout, and show how to colour the edges given fixed linear orderings of the vertex colour classes. We then describe methods for the manipulation of track layouts. For example, we show how to decrease the number of edge colours in a track layout at the expense of increasing the number of tracks, and vice versa. We then study the relationship between track layouts and other models of graph layout, namely stack and queue layouts, and geometric thickness. One of our principle results is that the queue-number and track-number of a graph are tied, in the sense that one is bounded by a function of the other. As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number. Finally we consider track layouts of planar graphs. While it is an open problem whether planar graphs have bounded track-number, we prove bounds on the track-number of outerplanar graphs, and give the best known lower bound on the track-number of planar graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as Technical Report TR-2003-07, School of Computer Science, Carleton University, Ottawa, Canad

Almost-equidistant sets

For a positive integer d, a set of points in d-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let f(d) denote the largest size of an almost-equidistant set in d-space. It is known that f(2) = 7 , f(3) = 10 , and that the extremal almost-equidistant sets are unique. We give independent, computer-assisted proofs of these statements. It is also known that f(5) ≥ 16. We further show that 12 ≤ f(4) ≤ 13 , f(5) ≤ 20 , 18 ≤ f(6) ≤ 26 , 20 ≤ f(7) ≤ 34 , and f(9) ≥ f(8) ≥ 24. Up to dimension 7, our work is based on various computer searches, and in dimensions 6–9, we give constructions based on the known construction for d= 5. For every dimension d≥ 3 , we give an example of an almost-equidistant set of 2 d+ 4 points in the d-space and we prove the asymptotic upper bound f(d) ≤ O(d 3 / 2)

Canonical Tverberg partitions.

We show that for every $d,r,N$ positive integers there exists $n = n(d,r,N)$ such that Any sequence $p$ in $R^d$ of length $n$ has a subsequence $pâ$ of length $N$ such that Every subsequence of $pâ$ of length $T(d,r) = (r-1)(d+1)+1$ has identical Tverberg partitions, namely the ârainbowâ â partitions. A partition (or coloring) of the first $T(d,r)$ integers into $r$ parts (with $r$ colors) is called rainbow If every color appears exactly once in each of the following $r$-tuples: $(1, ⠦., r), (r, ⠦, 2r-1), (2r-1, ⠦, 3r-2), ⠦., ((d-1)r-(d-2), ⠦, d r-(d-1))$.Non UBCUnreviewedAuthor affiliation: Western Kentucky UniversityFacult

maximizing maximal angles for plane straight-line graphs

Let G = (S, E) be a plane straight-line graph on a finite point set S subset of R-2 in general position. The incident angles of a point p is an element of S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called phi-open if each vertex has an incident angle of size at least phi. In this paper we study the following type of question: What is the maximum angle phi such that for any finite set S subset of R-2 of points in general position we can find a graph from a certain class of graphs on S that is phi-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases. (C) 2012 Elsevier B.V. All rights reserved.Austrian Science Fund (FWF), NRN 'Industrial Geometry S9205-N12; Austrian Science Fund (EWE) P23629-N18; project MEC MTM2009-07242; project DGR 2009SGR1040; Spanish Ministry of Science T60427, MTM2008-04699-C03-02, CSD2006-00032Let G = (S, E) be a plane straight-line graph on a finite point set S subset of R-2 in general position. The incident angles of a point p is an element of S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called phi-open if each vertex has an incident angle of size at least phi. In this paper we study the following type of question: What is the maximum angle phi such that for any finite set S subset of R-2 of points in general position we can find a graph from a certain class of graphs on S that is phi-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases. (C) 2012 Elsevier B.V. All rights reserved

Every large point set contains many collinear points or an empty pentagon

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]