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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), 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 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 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 b x 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 b x 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 b x z y 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 b x z y degenerate Manhattan lines a b x y z a b x y z a a b b x x y y z z a b x z y 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 ): 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)n 1/2 distinct lines or else some line consists of all these n points.