Optimal Point Sets Determining Few Distinct Angles

TL;DR

This paper characterizes the maximum size of planar point sets that determine at most a certain number of angles, establishing exact values for small cases and bounds for general cases, revealing a linear relationship.

Contribution

It provides exact maximum sizes for point sets with at most 2 or 3 angles and bounds for general k, linking geometric angle problems to combinatorial bounds.

Findings

P(2)=5 and P(3)=5 for the maximum size of point sets with limited angles

Established bounds: k+2 ≤ P(k) ≤ 6k for general k

Revealed a surprising linear relationship P(k)=Θ(k)

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 well-understood.