Canada Research Chairs Communication Guidelines for Academic Publications and Presentations

1. 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. 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.

3. 5 points, 5 lines nothing between these two 5 points, 1 line b 5 points 10 lines b 5 points 6 lines

4. 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)

5. Paul Erdős Nicolaas de Bruijn

6. 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.

7. 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?

8. a b b Observation z y x a y x z This can be taken for a definition of a line ab in an arbitrary metric space

9. Lines in metric spaces can be exotic One line can hide another!

10. C E D B B B C D B E A line BC consists of A,B,C,E line BD consists of B,D,E line BE consists of A,B,C,D,E E A E A,D,C

11. 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

12. 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)

13. Apart from the special graphs, we know only that In every connected graph on n vertices, there are at least n distinct lines or else some line consists of all n vertices. In every connected graph on n vertices, there are 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.

14. A variation (Yori Zwols, 2012): In every connected graph on n vertices, there are at least n distinct lines or else some line consists of all n vertices. 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 A graph theory conjecture:

15. Another partial result: 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 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. The general conjecture: Ida Kantor and Balász Patkós, this conference

16. Apart from the special cases, we know only that 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. The general conjecture: Xiaomin Chen and V.C., Problems related to a De Bruijn - Erdős theorem, Discrete Applied Mathematics 156 (2008), 2101 - 2108

17. LUNCH!!