A de Bruijn - Erd\H{o}s theorem and metric spaces (original) (raw)

A de Bruijn - Erdos theorem and metric spaces

Discrete Mathematics & Theoretical Computer Science

Combinatorics De Bruijn and Erdos proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.

A de Bruijn-Erdős theorem for 1–2 metric spaces

Czechoslovak Mathematical Journal, 2014

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.

Discrete Applied Mathematics, 2008

De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on related extremal combinatorial problems.

A De Bruijn-Erdos theorem for chordal graphs

A remark on finite metric spaces

arXiv (Cornell University), 2022

Richmond and Richmond (American Mathematical Monthly 104 (1997), 713-719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove that the hypothesis is unnecessarily strong: In a metric space on n points, n 3n + 5 arbitrarily placed or 3 n-2 2 + 1 suitably placed degenerate triangles suffice. 1 Results. Given a metric space (V, dist), we follow [1] in writing [rst] to signify that r, s, t are pairwise distinct points of V and dist(r, s) + dist(s, t) = dist(r, t). Following , we refer to three-point subsets of V as triangles; if [rst], then the triangle {r, s, t} is called degenerate. Now let (V, dist) be a metric space. Trivially, if there is a linear order on V such that r ≺ s ≺ t ⇒ [rst], then all triangles in V are degenerate. Richmond and Richmond [15] proved the converse under a mild lower bound on |V |:

De Bruijn–Erdős-type theorems for graphs and posets

Discrete Mathematics, 2017

A classical theorem of De Bruijn and Erdős asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to comparability graphs, we obtain a version of the De Bruijn-Erdős theorem for partially ordered sets (posets). Moreover, in this case, we have an improved bound on the number of lines depending on the height of the poset. The extremal configurations are also determined.

Erdős–Szekeres Theorem for Lines

Discrete & Computational Geometry, 2015

According to the Erdős-Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erdős-Szekeres theorem.

Lines in quasi-metric spaces with four points

Cornell University - arXiv, 2022

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvátal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. In this work we prove that there is a quasi-metric space on four points a, b, c and d whose betweenness is B = {(c, a, b), (a, b, c), (d, b, a), (b, a, d)}. Then, this space has only three lines none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we get that Chen and Chvátal's conjecture is valid for any metric space on four points.

Sylvester?Gallai Theorem and Metric Betweenness

Discrete and Computational Geometry, 2004

Sylvester conjectured in 1893 and Gallai proved some 40 years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the Sylvester-Gallai theorem generalizes as follows: in every finite metric space there is a line consisting of either two points or all the points of the space. Then we present meagre evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness.

On the Erdős Distance Conjecture in Geometry

Erdős asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1 1 i n ≤ ≤ −) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2 8 n ≤ ≤ [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4-7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.

The Hausdorff Metric Geometry REU 2005 Report

2005

The Hausdorff metric h gives us a method of measuring the distance between non-empty compact subsets of n-dimensional Euclidean space. Unlike the Euclidean concept of betweenness, there is not necessarily a unique set at each location between two other sets in the Hausdorff metric geometry. A configuration defines two sets (infinite or finite) for which it is possible to have a finite number of elements at each location between sets. We will first consider infinite and finite sets for which it is possible to have a finite number of sets at each location between two sets, noting that there is no configuration with exactly 19 elements at each location between two sets. We will conclude by connecting the number of sets at each location between two sets, making up specific finite configurations, and Fibonacci numbers.

Two results about points, lines and planes

Discrete Mathematics, 1986

Given n points in three dimensional euclidean space, not all lying on a plane, let 1 be the number of lines determined by the points, and let p be the number of planes determined. We show that 1'3 cnp, where c > 0. This is the weak version of the so-called Points-Lines-Planes conjecture (a conjecture of considerable interest to combinatorialists) being an instance of the conjectured log-concavity of the Whitney numbers. We also show that there is always a point incident with at least cl planes, where c > 0, provided that the n points do not all lie on two skew lines. This result lends support to our conjecture, published in 1981, that n -1 +p + 2 2 0.

A Higher Dimensional Version of a Problem of Erdős

2013

Let {p1, . . . , pn} ⊆ Rd. We think of d n. How big is the largest subset X of points such that all of the distances determined by elements of ( X 2 ) are different? We show that |X| ≥ Ω(n 1 3d−3+o(1) ). This improves on the best known result which was |X| ≥ Ω(n 1 3d−2 ). Assume that no a of the points are on the same (a− 1)-hyperplane. How big is the largest subset X of points such that all of the volumes determined by elements of ( X a ) are different? We show that |X| ≥ Ω(n 1 (2a−1)d ). This concept had not been studied before. Let α be a regular cardinal between א0 and 2א0 . Let X ⊆ Rd such that no a of the original points are in the same (a − 1)-hyperplane. We show that there is an α-sized subset of X such that all of the volumes determined by elements of ( X a ) are different. We give two proofs: one assuming the Axiom of Choice and one assuming the Axiom of Determinacy.

A Quantitative Version of the Erdős-Anning Theorem

2021

Let R ⊂ R n be an infinite set of collinear points and S ⊂ R be an arbitrary and finite set with S ⊂ N n . Then the number of points with mutual integer distances on the shortest line containing points in S satisfies the lower bound is the compression gap of the compression induced on x. This proves that there are infinitely many collinear points with mutual integer distances on any line in R n and generalizes the well-known Erdős-Anning Theorem in the plane R 2 .

A Further Extension of the KKMS Theorem

Mathematics of Operations Research, 2000

Recently Reny and Wooders ([23]) showed that there is some point in the intersection of sets in Shapley's ([24]) generalization of the Knaster-Kuratowski-Mazurkiwicz Theorem with the property that the collection of all sets containing that point is partnered as well as balanced. In this paper we provide a further extension by showing that the collection of all such sets can be chosen to be strictly balanced, implying the Reny-Wooders result. Our proof is topological, based on the Eilenberg-Montgomery¯xed point Theorem. Reny and Wooders ([23]) also show that if the collection of partnered points in the intersection is countable, then at least one of them is minimally partnered. Applying degree theory for correspondences, we show that if this collection is only assumed to be zero dimensional (or if the set of partnered and strictly balanced points is of dimension zero), then there is at least one strictly balanced and minimally partnered point in the intersection. The approach presented in this paper sheds a new geometric-topological light on the Reny-Wooders results.

On equidistant lines of given line configurations

Georgian Mathematical Journal, 2019

The equidistant set of a collection F of lines in 3-space is the set of those points whose distances to the lines in F are all equal. We present many examples and results related to the lines possibly contained in the equidistant set of F. In particular, we determine the possible numbers of lines in the equidistant set of a collection of n lines for every n > 0 {n>0} . For example, if n = 3 {n=3} , then the possible number of such lines is either 4 or 2 or 1 or 0. In a natural way, our results are connected with properties of special types of (ruled) surfaces. For example, we obtain also results on the number of lines in the intersection of quadratic surfaces.

On the dimension of n-point sets

Topology and its Applications, 2003

We give an affirmative answer to a question raised by Khalid Bouhjar and Jan J. Dijkstra concerning whether or not every one-dimensional partial n-point set contains an arc by showing that a partial n-point set is one-dimensional if and only if it contains an arc.