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

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 ${135}^{\circ }\le \theta <{180}^{\circ }$, 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.