Contaje de triángulos en conjuntos de puntos coloreados: un problema de la geometría combinatoria

A classical object of study in combinatorial geometry are sets S of points in the plane. A triangle with vertices from S is called empty if it contains no points of S in its interior. The number of empty triangles depends on the positions of points from S and a burning question is: How many empty triangles are there at least, among all sets S of n points? In order to discard degenerate point configurations, we only consider sets S without three collinear points. In this project, a software has been developed which allows to count the number of empty triangles in a set of n points in the plane. The software permits generation of point sets and their graphical visualization, as well as searching and displaying of optimal point configurations encountered. A point set of a given cardinality is said to be optimal if it contains the minimum number of empty triangles. The objective is to derive bounds on the minimum number of empty triangles by means of experiments realized with our software. The created program also allows to count empty monochromatic triangles in two-colored point sets. A triangle is called monochromatic if its three vertices have the same color. While the first problem has been studied extensively during the last decades, the two-colored version remains to be explored in depth. In this work we also expose our results on the minimum number of empty triangles in (small) two-colored point sets. Also, the treated problem is put in context with related results, such as the Erdös-Szekeres theorem, and a short outline of famous problems which contributed to the rise of combinatorial geometry is presented.Un objeto clásico de estudio en la Geometría combinatoria son conjuntos S de n puntos en el plano. Se dice que un triángulo con vértices en S esta vacío si no contiene puntos de S en su interior. El número de triángulos vacíos depende de cómo se dibujó el conjunto S y una pregunta ardiente es: ¿Cuántos triángulos vacíos hay como mínimo en cada conjunto S de n puntos? Para descartar configuraciones de puntos degeneradas solo se consideran nubes de puntos sin tres puntos colineales

Empty monochromatic simplices

Let S be a k-colored (finite) set of n points in , da parts per thousand yen3, in general position, that is, no (d+1) points of S lie in a common (d-1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3a parts per thousand currency signka parts per thousand currency signd we provide a lower bound of and strengthen this to Omega(n (d-2/3)) for k=2.; On the way we provide various results on triangulations of point sets in . In particular, for any constant dimension da parts per thousand yen3, we prove that every set of n points (n sufficiently large), in general position in , admits a triangulation with at least dn+Omega(logn) simplices.Postprint (author’s final draft

Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+\Omega(\log n)$ simplices

Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant

Coloring half-planes and bottomless rectangles

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms

Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $\chi_2$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere $S^3$ with face vector $f=(167,1579,2824,1412)$ and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

Almost Empty Monochromatic Triangles in Planar Point Sets

For positive integers c, s ≥ 1, let M3 (c, s) be the least integer such that any set of at least M3 (c, s) points in the plane, no three on a line and colored with c colors, contains a monochromatic triangle with at most s interior points. The case s = 0 , which corresponds to empty monochromatic triangles, has been studied extensively over the last few years. In particular, it is known that M3 (1, 0) = 3, M3 (2, 0) = 9, and M3 (c, 0) = ∞, for c ≥ 3. In this paper we extend these results when c ≥ 2 and s ≥ 1. We prove that the least integer λ3 (c) such that M3 (c, λ3 (c)) \u3c ∞ satisfies: ⌊(c-1)/2⌋ ≤ λ3 (c) ≤ c - 2, where c ≥ 2. Moreover, the exact values of M3 (c, s) are determined for small values of c and s. We also conjecture that λ3 (4) = 1, and verify it for sufficiently large Horton sets