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b

5 points

10 lines

5 points

6 lines

5 points, 5 lines

b

5 points, 1 line

nothing between these two

Every set of n points in the plane

determines at least n distinct lines unless

all these n points lie on a single line.

This is a corollary of the Sylvester-Gallai theorem

(Erdős 1943)

Nicolaas de Bruijn Paul Erdős

Every set of n points in the plane

determines at least n distinct lines unless

all these n points lie on a single line.

A generalization by de Bruijn and Erdős

On a combinatorial problem.Indag. Math. 10 (1948), 421--423

Every set of n points in the plane

determines at least n distinct lines unless

all these n points lie on a single line.

What other icebergs

could this theorem be a tip of?

a b

x y z

a bx y z

Observation

This can be taken for a definition of a line ab

in an arbitrary metric space

Lines in metric spaces can be exotic

One line can hide another!

B C

line BC consists of A,B,C,E

line BD consists of B,D,EB C

E

D

A

B D

B E

line BE consists of A,B,C,D,E

A E

A,D,C

E

Conjecture (Xiaomin Chen and V.C., 2006):

In every metric space on n points,

there are at least n distinct lines or else

some line consists of all n points.

In every connected graph on n vertices,

there are at least n distinct lines or else

some line consists of all n vertices.

Special case:

5 vertices, 4 lines

In every connected graph on n vertices,

there are at least n distinct lines or else

some line consists of all n vertices.

A graph theory conjecture:

True for special graphs:

Bipartite graphs (Exercise)

Chordal graphs (Laurent Beaudou, Adrian Bondy, Xiaomin Chen, Ehsan Chiniforooshan, Maria Chudnovsky, V.C., Nicolas Fraiman, Yori Zwols, A De Bruijn - Erdős theorem for chordal graphs, arXiv, 2012)

Graphs of diameter two (V.C., A De Bruijn - Erdős theorem for 1-2 metric spaces, arXiv, 2012)

In every connected graph on n vertices,

there are at least n distinct lines or else

some line consists of all n vertices.

A graph theory conjecture:

Ehsan Chiniforooshan and V.C., A De Bruijn - Erdős theorem and metric spaces,Discrete Mathematics & Theoretical Computer Science Vol 13 No 1 (2011), 67 - 74.

Apart from the special graphs, we know only that

In every connected graph on n vertices,

there are distinct lines or else

some line consists of all n vertices.

In every connected graph on n vertices,

there are at least n distinct lines or else

some line consists of all n vertices.

A graph theory conjecture:

A variation (Yori Zwols, 2012):

In every square-free connected graph on n vertices,

there are at least n distinct lines or else

the graph has a bridge.

4 vertices, 1 line, no bridge

The general conjecture:

In every metric space on n points,

there are at least n distinct lines or else

some line consists of all these n points.

Ida Kantor and Balász Patkós, this conference

In every L1-metric space on n points in the plane,

there are at least n/37 distinct lines or else

some line consists of all these n points.

Another partial result:

The general conjecture:

In every metric space on n points,

there are at least n distinct lines or else

some line consists of all these n points.

In every metric space on n points,

there are at least distinct lines or else

some line consists of all these n points.

Apart from the special cases, we know only that

Xiaomin Chen and V.C., Problems related to a De Bruijn - Erdős theorem,Discrete Applied Mathematics 156 (2008), 2101 - 2108

LUNCH!!