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14. Collinear triples and self-intersection paths

    1. Collinear triples and self-intersection paths

          For a set $P$ in the plane without collinear $4$-tuples, we define a geometric graph $G_P$ with vertices being $P$ and edges “being" its collinear triples: if $u,v,w$ is a collinear triple in consecutive order, then $uv$ and $vw$ are two edges drawn as straight-line segments.

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      Problem 14.1.

      [Joszef Solymosi] What is the minimal number of collinear triples in any set $P$ of $n$ points in order to guarantee a self-intersection $3$-path in $G_P$, i.e. a path $abcd$ with $ab$ and $cd$ crossing?

          Cite this as: AimPL: Albertson conjecture and related problems, available at http://aimpl.org/albertson.