On the sets of $n$ points forming $n+1$ directions
Abstract
Let $S$ be a set of $n\geq 7$ points in the plane, no three of which are collinear. Suppose that $S$ determines $n+1$ directions. That is to say, the segments whose endpoints are in $S$ form $n+1$ distinct slopes. We prove that $S$ is, up to an affine transformation, equal to $n$ of the vertices of a regular $(n+1)$gon. This result was conjectured in 1986 by R. E. Jamison. In an addendum to the paper, we show that a much stronger result can be obtained as a corollary of a structure theorem of Green and Tao on point sets spanning few ordinary lines.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 DOI:
 10.48550/arXiv.1811.01055
 arXiv:
 arXiv:1811.01055
 Bibcode:
 2018arXiv181101055P
 Keywords:

 Mathematics  Combinatorics;
 52C10;
 52C30;
 52C35
 EPrint:
 Paper: 7 pages, 5 figures. Addendum: 3 pages