Optimal Point Sets Determining Few Distinct Angles
Abstract
We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \leq P(k) \leq 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=\Theta(k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is wellunderstood.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.12034
 Bibcode:
 2021arXiv210812034F
 Keywords:

 Mathematics  Combinatorics;
 05