Canada Research Chairs In 2000, the Government of Canada created a permanent program to establish 2000 research professorships—Canada Research Chairs—in eligible degree-granting institutions across the country. Communication Guidelines for Chairholders In all professional publications, presentations and conferences, we ask you to identify yourself as a Canada Research Chair and acknowledge the contribution of the program to your research.
Canada Research Chairs In 2000, the Government of Canada created a permanent program to establish 2000 research professorships—Canada Research Chairs—in.
Dec 17, 2015
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Canada Research Chairs
In 2000, the Government of Canada created a permanent program to establish 2000 research professorships—Canada Research Chairs—in eligible degree-granting institutions across the country.
Communication Guidelines for Chairholders
In all professional publications, presentations and conferences, we ask you to identify yourself as a Canada Research Chair and acknowledge the contribution of the program to your research.
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 correct solution: Tibor Gallai (1933)
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?
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!
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.
Manhattan distance
a bx zy lines become
a
b
x
y
z Manhattan 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 zydegenerate Manhattan lines
a
b
x
y
z
a bxy
z
a
a b
b
x
x
yy
z
z
a bx zy typical Manhattan lines
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.
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