A De Bruijn-Erd\H{o}s theorem in graphs?

TL;DR

This paper explores a De Bruijn-Erdős type theorem in general metric spaces, focusing on graphs, and discusses related open problems and conjectures in the field.

Contribution

It investigates the validity of a De Bruijn-Erdős theorem in metric spaces derived from graphs and highlights numerous open problems and conjectures.

Findings

The theorem remains unresolved in graph metric spaces.

Identifies 29 open problems related to the theorem.

Proposes 3 new conjectures in the area.

Abstract

A set of $n$ points in the Euclidean plane determines at least $n$ distinct lines unless these $n$ points are collinear. In 2006, Chen and Chv\'atal asked whether the same statement holds true in general metric spaces, where the line determined by points $x$ and $y$ is defined as the set consisting of $x$, $y$, and all points $z$ such that one of the three points $x,y,z$ lies between the other two. The conjecture that it does hold true remains unresolved even in the special case where the metric space arises from a connected undirected graph with unit lengths assigned to edges. We trace its curriculum vitae and point out twenty-nine related open problems plus three additional conjectures.

Tables

Table 1. Table 1: Comparison of the two proofs

Euclidean plane: Kelly	General metric space: Chen
\svhline 1. If some three points of $V$ are noncollinear,	1A. If some three points are in no closure line,
then some line passes through only two points of $V$:xxx	then some simple triangle is in no closure line:
	if $a,b,c$ minimize
	$dist(a,b)+dist(b,c)+dist(a,c)$
	over all triples of points in no closure line,
	then $\{a,b,c\}$ is a simple triangle.
	1B. If some simple triangle is in no closure line,
	then some closure line consists of two points:
if $a,b,c$ minimize	if $a,b,c$ minimize
the distance of point $b$ from line $\overline{ac}$,	$dist(a,b)+dist(b,c)-dist(a,c)$
over all noncollinear triples,	over all simple triangles,
then $\overline{ac}$ passes through no third point of $V$.	then $C\left(ac\right)=\{a,c\}$.
2. If every three points of $V$ are collinear,	2. If every three points are in some closure line,
then all points of $V$ are collinear.	then some closure line consists of all the points.

Equations

$$\left(\genfrac{}{}{0ex}{}{k}{2}\right)+{\sum }_{i=1}^k\left(\genfrac{}{}{0ex}{}{n_i}{2}\right)=\left(\genfrac{}{}{0ex}{}{k}{2}\right)+{\sum }_{i=m+1}^k\left(\genfrac{}{}{0ex}{}{n_i}{2}\right)\ge k+{\sum }_{i=m+1}^k{n}_i=k+(n-m)\ge n.$$

$$I(a,b)$$

$$O(a,b)$$

$$(u,v,w)\in B(M)\;\Leftrightarrow\;\mbox{ $u,v,w$ are all distinct and $dist(u,v)+dist(v,w)=dist(u,w)$ }$$

$$L(xy)=\{x,y\}\cup \{z:(x,y,z)\in B(M)\vee (y,z,x)\in B(M)\vee (z,x,y)\in B(M)\}.$$

$$T(B)=\{\{u,v,w\}:(u,v,w)\in B\}.$$

$$L(xy)=\{x,y\}\cup \{z:\{x,y,z\}\in T(B)\}.$$

$$dist(a,b)=dist(b,c)=1,dist(a,c)=dist(a,d)=dist(b,d)=dist(c,d)=2$$

Full text

1 Prehistory

It all started when Alain Guenoche and Bernard Fichet asked me if I wanted to come to their third International Conference on Discrete Metric Spaces in Marseilles in September 1998. I like metric spaces and I love Marseilles, I replied, but I have no results I could present there. Never mind, they said magnanimously, come anyway. As I am not completely without shame, I then began racking my brain for something to talk about at the conference. A distant memory came to the rescue: As an undergraduate, I marvelled at the interpretation of families of sets as metric spaces provided by the Hamming metric on a family of indicator functions. Could a few combinatorial theorems be generalized to the realm of metric spaces? Dusting off my youthful ambition thirty years later, I circled around it till I settled on the project of looking for theorems of Euclidean geometry that might be generalized to arbitrary metric spaces.

1.1 Lines and closure lines in metric spaces

Saying that point $v$ in a Euclidean space lies between points $u$ and $w$ means that $v$ is an interior point of the line segment with endpoints $u$ and $w$; line $\overline{xy}$ is the set consisting of $x$, $y$, and all points $z$ such that one of the three points $x,y,z$ lies between the other two. These notions have straightforward extensions to arbitrary metric spaces: In a space with metric $dist$, saying that point $v$ lies between points $u$ and $w$ means that $u,v,w$ are pairwise distinct and $dist(u,v)+dist(v,w)=dist(u,w)$; if line $L(xy)$ is defined as the set consisting of $x$, $y$, and all points $z$ such that one of the three points $x,y,z$ lies between the other two, then $L(xy)=\overline{xy}$ in the special case where $dist$ is the Euclidean metric.

This was the definition of lines in metric spaces that I hoped to use in extending a theorem or two of Euclidean geometry to arbitrary metric spaces. One candidate was the Sylvester–Gallai theorem Syl ; E82 ,

• Every non-collinear finite subset $V$ of the Euclidean plane such that $\lvert V\rvert\geq 2$

includes two points such that the line determined by them passes through

no other point of $V$,

whose generalization would read

• In every finite metric space $(V,dist)$ such that $\lvert V\rvert\geq 2$, some line consists of only two points of $V$ or of all points of $V$.

This candidate flunked miserably: When $(V,dist)$ is the pentagon $C_{5}$ with the usual graph metric (in this case, $dist(x,y)=1$ when vertices $x,y$ are adjacent and $dist(x,y)=2$ when vertices $x,y$ are nonadjacent), $L(xy)$ consists of four vertices when $x,y$ are adjacent and it consists of three vertices when $x,y$ are nonadjacent.

Undaunted, I tried another tack: Let us define closure line $C(xy)$ as the smallest superset of $L(xy)$ such that $u,v\in C(xy)\Rightarrow L(uv)\subseteq C(xy)$. Just like lines $L(xy)$, closure lines $C(xy)$ are identical with Euclidean lines $\overline{xy}$ in the special case where the metric is Euclidean. Unlike lines $L(xy)$, closure lines $C(xy)$ did not flunk the Sylvester–Gallai test at once: I could not find a counterexample to the statement

• (SG) In every finite metric space $(V,dist)$ such that $\lvert V\rvert\geq 2$, some closure line consists of only two points of $V$ or of all points of $V$.

(In particular, $C_{5}$ is not a counterexample as each of its ten closure lines consists of all five vertices.)

1.2 Sylvester–Gallai theorem in metric spaces?

Having formulated generalization (SG) of the Sylvester–Gallai theorem, I tried to prove it. The known proofs of the Sylvester–Gallai theorem E43 ; C48 ; C61 did not help: I failed to adapt any of them to a proof of (SG). I considered the restricted version of (SG) where the metric spaces are induced by graphs: every connected undirected graph with vertex set $V$ induces the metric space $(V,dist)$ where $dist$ is the usual graph metric ($dist(u,v)$ standing for the number of edges in the shortest path from $u$ to $v$). This turned out to be easy: not only the restricted version of (SG), but even a stronger statement,

• In every finite metric space $(V,dist)$ induced by a graph with at least two vertices, every closure line consists of only two points of $V$ or of all points of $V$,

is valid. (The proof is a simple exercise: if $x$ and $y$ are adjacent twins, then $C(xy)=\{x,y\}$; else $C(xy)=V$.) To get more faith in the validity of (SG), I then tried to show that a counterexample could not be ridiculously small; plodding case analysis aided by computer search established that (SG) holds true for all metric spaces with at most nine points. Armed with this pathetic evidence, I presented the arrogant conjecture and related observations Ch04 at the Marseilles meeting.

Over the next few years, I publicized the conjecture vigorously. I told it to anybody who would listen. I told it to first-class researchers and some of them may have taken a crack at it. I gave talks about it in different places. A mathematical luminary interrupted my lecture at Princeton to announce that he had a counterexample; a few minutes later he and the entire audience agreed that the example was not a counterexample. After the lecture; he proposed to me (now privately) a new counterexample; this, too, turned out to be false. Such episodes made me feel that the conjecture may have been not all that arrogant.

The conjecture remained unresolved till the fall of 2003.

1.3 Enter Xiaomin

It was March 2000. There was a knock and when I opened my office door, there stood a young man who asked for a few minutes of my time. He explained to me his personal reasons for wanting to come to Rutgers as a graduate student in the middle of spring term and asked me if I could help by putting in a good word for him.

I said I sympathized, but as I didn’t know him from Adam, I could not put in a good word for him. He replied that he anticipated this reaction and perhaps I could give him a test to get an idea of his mathematical abilities? As I was just about to leave for my graduate class in algorithms and data structures, I handed to him a copy of the midterm exam I was going to give in a few minutes and asked him to come back after class. He looked the exam over, asked for definitions of a couple of concepts he was unfamiliar with, and then we went our separate ways. When I returned and read his answers, my jaw dropped: they were a notch above those of the thirty students who had studied the material for half a term. It was only later and after much prodding from me that he reluctantly confessed to his high ranking in the Chinese Mathematical Olympiad. (China being a biggish country, I was much impressed, of course.)

I gave him a glowing recommendation, he was admitted, and the rest is history. His name was Xiaomin Chen.

1.4 Sylvester–Gallai theorem in metric spaces!

In the fall of 2003, Xiaomin proved conjecture (SG).

A pivotal notion in Leroy Milton Kelly’s celebrated short proof C48 , (C61, , Section 4.7), (G12, , Chapter 8) of the Sylvester–Gallai theorem is the distance of a point from a line. This notion is unavailable in general metric spaces and yet echoes of Kelly’s proof can be found in Chen’s. Kelly minimizes the distance of point $b$ from line $\overline{ac}$ over all noncollinear triples $a,b,c$, which can be seen as choosing the flattest triangle with base $ac$ and apex $b$; Chen minimizes $dist(a,b)+dist(b,c)-dist(a,c)$, which can also be seen as choosing the flattest triangle with base $ac$ and apex $b$. Here, the following definitions are required to overcome a technical wrinkle: in a metric space:

• •

a triangle is a set of three points, none of which lies between the other two;

• •

its three edges are its two-point subsets;

• •

an edge is simple if no point lies between its two points;

• •

a triangle is simple if all three of its edges are simple.

Synopses of the two proofs are compared in Table 1.4.