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

Probability Level 4

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

Find least possible value of nn, such that we can always choose 2 points out of nn 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)(x,y), where x,yRx,y \in \mathbb{R}

\bullet Lattice points are points that have integer coordinates.


Harder versions 3D and 5D

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