Optimal point sets determining few distinct angles

Let P(k) denote the largest size of a non-collinear point set in the plane admitting at most k distinct angles. We prove P (2) = P (3) = 5, and we characterize the optimal sets. We also leverage results from Fleischmann et al. [Disc. Comput. Geom. (2023)] to provide the general bounds k+2 < P(k) < 6k, although the upper bound may be improved pending progress toward the Strong Dirac Conjecture. We conjecture that the lower bound is tight, providing infinite families of configurations meeting the bound and ruling out several classes of potential counterexamples.