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I am looking for a solution of the No Three-in-a-Line problem for the whole $\mathbb{Z}\times\mathbb{Z}$ plane and had the idea, to use a non-redundant enumeration of the rationals, like the breadth-first enumeration of the knots in the Stern-Brocot tree or the Calkin-Wilf tree.

Question:

if an infinite sequence of points in $(x_i,y_i)\in\mathbb{Z}\times\mathbb{Z}$ corresponds to the sequence of rationals generated by one of their enumerations $\frac{x_i}{y_i}$ (as mentioned above), which of those points $(x_k,y_k)$ have the property, that they are

  • collinear with $(x_i,y_i)$ and $(x_j,y_j)$, where $i\lt j\lt k$

  • the ones, that are collinear with exactly one of the pairs $(x_i,y_i)$ and $(x_j,y_j)$ for $i\lt j\lt k$?

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  • $\begingroup$ And what is the "No three in a LIne" Problem on the entire $\mathbb Z\times\mathbb Z$ plane? $\endgroup$
    – fedja
    Commented Dec 13, 2017 at 1:39
  • $\begingroup$ @fedja it is a maximal set of points with integer coordinates, that are not collinear with a pair of different points, that both have strictly smaller distance to the origin. $\endgroup$
    – Manfred Weis
    Commented Dec 13, 2017 at 7:47

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