2021/08/26 by Fleischmann, Henry L., Miller, Steven J., Palsson, Eyvindur A. +2
#05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2108.12034
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 ≤ P(k) ≤ 6k, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that P(k)=Θ(k) since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.