The extensible No-Three-In-Line problem
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866917829266112512 |
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| author | Nagy, Dániel T. Nagy, Zoltán Lóránt Woodroofe, Russ |
| author_facet | Nagy, Dániel T. Nagy, Zoltán Lóránt Woodroofe, Russ |
| contents | 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 $Θ(n/\log^{1+\varepsilon}n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_01447 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The extensible No-Three-In-Line problem Nagy, Dániel T. Nagy, Zoltán Lóránt Woodroofe, Russ Combinatorics 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 $Θ(n/\log^{1+\varepsilon}n)$ 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. |
| title | The extensible No-Three-In-Line problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2209.01447 |