A De Bruijn-Erdős Theorem for Asteroidal-Triple Free Graphs∗

A classical result of De Bruijn and Erdős shows that any given n points are either collinear or determine at least n distinct lines in the plane. Chen and Chvátal conjectured that a generalization of this result in all metric spaces, with appropriate definitions of lines, holds as well. In this note, we show that the Chen-Chvátal generalization holds for Asteroidal-Triple free graphs, where lines are defined using the notion of betweenness that rises naturally from the underlying convex geometry defined by this graph class. The Sylverster-Gallai theorem asserts that any set of n points is either collinear or there is a line that goes through exactly two distinct points. One of the corollaries of this result is the De Bruijn-Erdős theorem; which asserts that any n collinear points in the plane determine at least n distinct lines [10]. This in fact holds in settings where distances and angles are not needed; just the notions of lines and points on lines in ordered geometries. In particular, there is a close relationship between ordered geometries and the notion of betweenness defined by Menger [14]. Betweenness is a ternary relation, that relates the “placement” of a point x between two other points p and q. We write [pxq] to say that x lies between p and q, and write (E,B) to denote the set system defined on the ground set E of points, and a betweenness relation B defined on the elements of E. In this setting, one can define a line pq between any two distinct points p and q as follows: pq = {p, q} ∪ {x : [xpq] ∈ B ∨ [pxq] ∈ B ∨ [pqx] ∈ B} (1) Thus the definition of a line varies with every notion of betweenness. For instance, in Euclidean space, betweenness translates naturally to the Euclidean metric: [pxq] ⇐⇒ p, q, x are distinct points and dist(p, x) + dist(x, q) = dist(p, q), However in arbitrary spaces, metric betweenness leads to other types of lines. In fact, it can lead to families of lines with different behaviour. For instance, a line can be properly contained in another line, as shown in the example below [6]. Given dist(u, v) = dist(v, x) = dist(x, y) = dist(y, z) = dist(z, u) = 1 dist(u, x) = dist(v, y) = dist(x, z) = dist(y, u) = dist(z, v) = 2, the line vy = {v, x, y} is properly contained in the line xy = {v, x, y, z}. Chen and Chvátal conjecture in [6] the following Conjecture 0.1. Every metric space on n ≥ 2 points either has at least n distinct lines or has a line that contains all the points. Clearly, the definition of a line varies with that of betweenness. Let’s call a line universal if it contains all the points. And let’s say that a metric space satisfies the De Bruijn-Erdős property if it either has a universal line or at least n distinct lines. Conjecture 0.1 has been proven for special cases where the notion of betweenness is well understood. In particular, it was shown that the conjecture holds for posets and chordal graphs, and other graph classes, see for instance [1, 2, 8]. For many of these cases, the notion of betweenness rises naturally from ∗This work was supported by NSERC