论文标题
可扩展的无三重问题
The extensible No-Three-In-Line problem
论文作者
论文摘要
经典的不三合一问题寻求从$ n \ times n $网格中选择的最大点数，同时避免了界线三重。众所周知，最大值是$ n $的线性。在一个ERDE问题之后，我们试图从无限网格$ z^{2} $中选择一组大密度，同时避免了界线三重。我们显示了包含$θ（n/\ log^{1+ \ varepsilon} n）$ points $ [1，n]^{2} $中的$θ（n/\ log^{1+ \ varepsilon} n）的存在的存在。我们还提供了计算证据，表明可能存在一组晶格点，每块足够大的$ n \ times n $网格都至少具有$ n/2 $的点。
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.
