The extensible No-Three-In-Line problem

The classical No-Three-In-Line asks for the largest set of lattice points in the n by n grid that lie in general position; that is, so that no three points are on a common line.  It is well known that for any n, the largest such set is between n and 2n.  Erde asked for an infinite set of points in general position, having many points on an n by n grid (for every n).  In recent work joint with Dáni Nagy and Zoli Nagy, we have produced an infinite set with Theta(n/log^(1+ε)) points when intersected with each n by n grid, and suggested a construction that might give exactly n/2 points on such grids.

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