For k≥3 and sufficiently large n, the maximum number of points in an n×n grid with no k+1 collinear is exactly kn.
Primitive points in rational polygons.Canad
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No-$(k+1)$-in-line problem for $k \geqslant 3$
For k≥3 and sufficiently large n, the maximum number of points in an n×n grid with no k+1 collinear is exactly kn.