The extensible No-Three-In-Line problem

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The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an n×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 Z2 while avoiding a collinear triple. We show the existence of such a set which contains Θ(n/log1+ɛn) points in [1,n]2 for all n, where ɛ>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×n grid.