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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),

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 b x 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,E B 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), 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),

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