Lines, Betweenness and Metric Spaces

Abstract

A classic theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space.We prove that in any metric space with n points, either there is a line containing all the points or there are at least Ω( √ n) lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove a corresponding bound of Ω(n 2/5 ) lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are Ω(n 4/7 ) lines, improving the previous Ω(n 2/7 ) bound. We also prove that the number of lines in an n-point metric space is at least n/5w, where w is the number of different distances in the space, and we give an Ω(n 4/3 ) lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from {1, 2, 3}.

A De Bruijn-Erdős Theorem in Graphs?

“…All graphs featured in Theorem 8 are HHDfree, but not all HHD-free graphs can be constructed as in Theorem 8. For example, start with the C 4 that has vertices v 1 , v 2 , v 3 , v 4…”

“…With similar persistence, Adrian Bondy liked to point out again and again in our ConCoCO discussions that our progress would get a great boost if we understood which equivalence relations ≡ on the edge sets of K n arise from graphs with n vertices (or, more generally, from metric spaces on n points) in the sense that ab ≡ xy ⇔ L(ab) = L(xy). Section 6 of [3] contains results on distinct pairs of vertices that define the same line. In its notation (Definition 6.…”

“…As noted in[13, Theorem 3], its lower bound is best possible: complete multipartite graphs with Θ (|G| 2/3 ) parts of sizes in Θ (|G| 1/3 ) have Θ (|G| 4/3 ) distinct lines and no universal line. There are many other graphs with these properties: Example 7.8 in[3] exhibits arbitrarily large graphs G with Θ (|G| 4/3 ) distinct lines, no universal line, and unbounded diameter. A neat variation on this theme has been pointed out by Xiaomin Chen: each graph in his class consists of Θ (n 2/3 ) chordless cycles of lengths in Θ (n 1/3 ) that, apart from a vertex common to all of them, are pairwise vertex-disjoint.Conjecture 3 is known to be valid for graphs of bounded diameter: here is a more general result.Theorem 10.…”

A New Class of Graphs That Satisfies the Chen‐Chvátal Conjecture

“…The best-known lower bound for the number of lines in metric spaces with no universal line is ( √ n) [2]. As it is explained in [3], it suffices to prove Conjecture 1.1 for metric spaces with integral distances.…”

“…Any finite connected graph induces a metric space on its vertex set, where the distance between two vertices u and v is defined as the length of a shortest path linking u and v. Such metric spaces are called graph metrics and are the subject of this article. The bestknown lower bound on the number of lines in a graph metric with no universal line is (n 4/7 ) [2]. In [5] and [3] it is proved that Conjecture 1.1 holds for chordal graphs and for distance-hereditary graphs respectively.…”