The extensible No-Three-In-Line problem
| Autor: | Nagy, Dániel T., Nagy, Zoltán Lóránt, Woodroofe, Russ |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | European J. Combin. 114 (2023), Paper 103796, 11 pages |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.ejc.2023.103796 |
| Popis: | The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek to select sets of large density from the infinite grid $Z^{2}$ while avoiding a collinear triple. We show the existence of such a set which contains $\Theta(n/\log^{1+\varepsilon}n)$ points in $[1,n]^{2}$ for all $n$, where $\varepsilon>0$ is an arbitrarily small real number. We also give computational evidence suggesting that a set of lattice points may exist that has at least $n/2$ points on every large enough $n\times n$ grid. Comment: 12 pages, 3 figures |
| Databáze: | arXiv |
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