No-Three-in-a-$\Theta:$ Variations on the No-Three-in-a-Line Problem
| Autor: | Dodson, Natalie, Godbole, Anant, Gonzalez, Dashleen, Lynch, Ryan, Southern, Lani |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 15 pages, 16 figures |
| Databáze: | arXiv |
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