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Canada Research Chairs. 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. - PowerPoint PPT Presentation
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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.
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!!
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