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

Discrete Mathematics Level 4

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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