This plane can't crash, it doesn't fly

There are \(n\) distinct lattice points marked on the 2D plane (2 dimensional).

Find least possible value of \(n\), such that we can always choose 2 points out of \(n\) points (wherever they may be marked), such that there's at least one more lattice point on the segment joining them.


Details and assumptions:-

\(\bullet\) In the 2D plane (or Cartesian plane), every point can be represented as coordinates \((x,y)\), where \(x,y \in \mathbb{R}\)

\(\bullet\) Lattice points are points that have integer coordinates.


Harder versions 3D and 5D

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