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arXiv · 2311.13183

No-Three-in-a-$\Theta:$ Variations on the No-Three-in-a-Line Problem

Natalie Dodson·Anant Godbole·Dashleen Gonzalez·Ryan Lynch·Lani Southern
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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 $\theta$? 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 $\theta$ such that $135^\circ \leq \theta < 180^\circ$, and further explore the case $\theta = 135^\circ$, 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 $\theta = 135^\circ$ to provide a general upper bound for all angles.

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Natalie Dodson, Anant Godbole, Dashleen Gonzalez, Ryan Lynch, Lani Southern. 2023-11-22. No-Three-in-a-$\Theta:$ Variations on the No-Three-in-a-Line Problem. https://arxiv.org/abs/2311.13183

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