Canada Research Chairs

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. ORDINARY LINES EXTRAORDINARY LINES?

4. James JosephSylvester Educational Times, March 1893 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, May 1893 H.J. Woodall, A.R.C.S. A four-line solution … containing two distinct flaws First correct solution: Tibor Gallai (1933)

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

6. 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

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. apply induction hypothesis to the remaining n-1 points This is a corollary of the Sylvester-Gallai theorem (Erdős 1943): remove this point

8. Combinatorial generalization Paul Erdős Nicolaas de Bruijn On a combinatorial problem, Indag. Math. 10(1948), 421--423 Let V be a finite set and let E be a family of of proper subsets of V such that every 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 Vif and only if E is either a near-pencil or else the family of lines in a projective plane.

9. 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?

10. A B E C D dist(A,B) = 1, dist(A,C) = 2, etc.

11. a b b 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). z y x a y x z This can be taken for a definition of a line L(ab) in an arbitrary metric space

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

13. b A B E z y x a C D L(AB) = {E,A,B,C} L(AC) = {A,B,C} One line can hide another!

14. 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.

15. x z y a b become lines z Manhattan lines y b a x Manhattan distance

16. 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”.

17. x x z z y y a a b b x z a y y b a b x z typical Manhattan lines degenerate Manhattan lines z b y y x z a b a x

18. 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. What if degenerate sets are allowed?

19. 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. Question (Chen and C. 2006):

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

21. bipartite not chordal not distance-hereditary chordal not bipartite not distance-hereditary distance-hereditary not bipartite not chordal

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