ORDINARY LINES

2. ORDINARY LINES EXTRAORDINARY LINES?

3. 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

4. First proof: T.Gallai (1933) L.M. Kelly’s proof: starting point far new line new point near starting line

5. Be wise: Generalize! or What iceberg is the Sylvester-Gallai theorem a tip of?

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

7. 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

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

9. 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!

10. b A B E z y x a C D L(AB) = {E,A,B,C} L(AC) = {A,B,C} no line consists of all points no line consists of two points

11. A B E C D If at first you don’t succeed, … 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) Observation: In metric subspaces of Euclidean spaces, C(ab)= L(ab). 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}

12. 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. Conjecture (V.C. 1998)

13. 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. Easy observation Lemma 2: If some three points are contained in no closure line, then some closure line consists of two points. Two-step proof Definitions: simple edge = two points such that no third point is between them simple triangle = abc such that ab,bc,ca are simple edges Step 1: If some three points are contained in no closure line, then some simple triangle is contained in no closure line. (minimize dist(a,b) + dist(b,c) + dist(a,c) over all such triples) 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 triangles; then closure line C(ac) equals {a,c})

15. 5 points, 5 lines nothing between these two 5 points, 1 line b 5 points 10 lines b 5 points 6 lines

16. 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

17. 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

18. Combinatorial generalization Paul Erdős Nicolaas de Bruijn 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. On a combinatorial problem, Indag. Math. 10(1948), 421--423

19. 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?

20. 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. “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.

21. Manhattan distance z y x a b becomes z y b a x

22. With Manhattan distance, precisely seven closure lines

24. 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”.

25. typical Manhattan lines: x z a y y b a b x z degenerate Manhattan lines: z b y y x z a b a x x z y a b

26. 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?

27. 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):

28. 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.

29. bipartite not chordal not distance-hereditary chordal not bipartite not distance-hereditary distance-hereditary not bipartite not chordal

30. 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 ):