Year: 2023
Type: preprint
Abstract: We pose a natural generalization to the well-studied and difficult no-three-in-a-line problem: How many points can be chosen on an $n \times n$ grid such that no three of them form an angle of $θ$? In this paper, we classify which angles yield nontrivial problems, noting that some angles appear in surprising configurations on the grid. We prove a lower bound of $2n$ points for angles $θ$ such that $135^\circ \leq θ< 180^\circ$, and further explore the case $θ= 135^\circ$, utilizing geometric properties of the grid to prove an upper bound of $3n - 2$ points. Lastly, we generalize the proof strategy used in proving the upper bound for $θ= 135^\circ$ to provide a general upper bound for all angles.
Source: arXiv (Cornell University)
Authors: Dodson, Natalie, Godbole, Anant, Gonzalez, Dashleen, Lynch, Ryan, Southern, Lani
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4388963499