ORDINARY LINES

EXTRAORDINARY LINES?

James Joseph Sylvester

Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line.

Educational Times, March 1893

Educational Times, May 1893

H.J. Woodall, A.R.C.S.

A four-line solution

… containing two distinct flaws

First proof: T.Gallai (1933)

L.M. Kelly’s proof:

starting line

starting point

new line

new pointnear

far

Be wise: Generalize!

or

What iceberg

is the Sylvester-Gallai theorem a tip of?

A

B

C

E

D

dist(A,B) = 1,

dist(A,C) = 2,

etc.

a b

x y z

a bx y z

This can be taken for a definition of a line L(ab)

in an arbitrary metric space

Observation

Line ab consists of all points x such that dist(x,a)+dist(a,b)=dist(x,b), all points y such that dist(a,y)+dist(y,b)=dist(a,b), all points z such that dist(a,b)+dist(b,z)=dist(a,z).

Lines in metric spaces can be exotic

One line can hide another!

a bx y z

A

B

C

E

D

L(AB) = {E,A,B,C}

L(AC) = {A,B,C}

One line can hide another!

a bx y z

A

B

C

E

D

L(AB) = {E,A,B,C}

L(AC) = {A,B,C}

no line consists of all points

no line consists of two points

Definition: closure line C(ab) is the smallest set S such that * a and b belong to S, * if u and v belong to S, then S contains line L(uv)

A

B

C

E

D

L(AB) = {E,A,B,C}

L(EA) = {D,E,A,B}

C(AB) = {A,B,C,D,E}

L(AC) = {A,B,C}

C(AC) = {A,B,C,D,E}

Observation: In metric subspaces of Euclidean spaces, C(ab) = L(ab).

If at first you don’t succeed, …

Conjecture (V.C. 1998)

Theorem (Xiaomin Chen 2003)

In every finite metric space,

some closure line consists of two points or else

some closure line consists of all the points.

The scheme of Xiaomin Chen’s proof

Lemma 1: If every three points are contained in some closure line, then some closure line consists of all the points.

Lemma 2: If some three points are contained in no closure line, then some closure line consists of two points.

Definitions: simple edge = two points such that no third point is between them simple triangle = abc such that ab,bc,ca are simple edges

Easy observation

Two-step proof

Step 1: If some three points are contained in no closure line, then some simple triangle is contained in no closure line.

Step 2: If some simple triangle is contained in no closure line, then some closure line consists of two points.

(minimize dist(a,b) + dist(b,c) + dist(a,c) over all such triples)

(minimize dist(a,b) + dist(b,c) - dist(a,c) over all such triangles; then closure line C(ac) equals {a,c})

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.

near-pencil

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):

remove this point

apply induction hypothesis to the remaining n-1 points

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

Combinatorial generalization

Nicolaas de Bruijn Paul Erdős

Let V be a finite set and let E be a family of of proper subsets of V such thatevery two distinct points of V belong to precisely one member of E.Then the size of E is at least the size of V. Furthermore, the size of E equals the size of V if and only if E is either a near-pencil or else the family of lines in a projective plane.

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?

“Closure lines” in place of “lines” do not work here:

For arbitrarily large n, there are metric spaces on n points,

where there are precisely seven distinct closure lines

and none of them consist of all the n points.

Question (Chen and C. 2006):

True or false? In every metric space on n points,

there are at least n distinct lines or else

some line consists of all these n points.

z

x

y

Manhattan distance

b

a

a bx y z

becomes

With Manhattan distance, precisely seven closure lines

Question (Chen and C. 2006):

True or false? In every metric space on n points,

there are at least n distinct lines or else

some line consists of all these n points.

Partial answer (Ida Kantor and Balász Patkós 2012 ):

Every nondegenerate set of n points in the plane

determines at least n distinct Manhattan lines or else

one of its Manhattan lines consists of all these n points.

“nondegenerate” means “no two points share their x-coordinate or y-coordinate”.

a bx zy

degenerate Manhattan lines:

aa

b

b

x

x yy

z

z

typical Manhattan lines:

a

a b

b

x

x

yy

z

z

What if degenerate sets are allowed?

Theorem (Ida Kantor and Balász Patkós 2012 ):

Every set of n points in the plane

determines at least n/37 distinct Manhattan lines or else

one of its Manhattan lines consists of all these n points.

Question (Chen and C. 2006):

True or false? In every metric space on n points,

there are at least n distinct lines or else

some line consists of all these n points.

Another partial answer (C. 2012 ):

In every metric space on n points

where all distances are 0, 1, or 2,

there are at least n distinct lines or else

some line consists of all these n points.

Another partial answer (easy exercise):

In every metric space on n points

induced by a connected bipartite graph,

some line consists of all these n points.

In every metric space on n points

induced by a connected chordal graph,

there are at least n distinct lines or else

some line consists of all these n points.

Another partial answer (Laurent Beaudou, Adrian Bondy, Xiaomin Chen, Ehsan Chiniforooshan, Maria Chudnovsky, V.C., Nicolas Fraiman, Yori Zwols 2012):

Another partial answer (Pierre Aboulker and Rohan Kapadia 2014):

In every metric space on n points

induced by a connected distance-hereditary graph,

there are at least n distinct lines or else

some line consists of all these n points.

bipartite not chordal not distance-hereditary

chordal not bipartite not distance-hereditary

distance-hereditary not bipartite not chordal

Theorem (Pierre Aboulker, Xiaomin Chen, Guangda Huzhang, Rohan Kapadia, Cathryn Supko 2014 ):

In every metric space on n points,

there are at least (1/3)n1/2 distinct lines or else

some line consists of all these n points.