Not many, just 5 dimensions

Probability Level 2

Find least possible value of nn, such that we can always choose 2 points out of nn points in 5-dimensional space (wherever they may be marked), such that there's at least one more lattice point on the segment joining them.

Details and assumptions:-

  • In the 5D space, every point can be represented as coordinates (a,b,c,d,e)(a,b,c,d,e), where a,b,c,d,e,fRa,b,c,d,e,f \in \mathbb{R}
  • Lattice points are points that have integer coordinates.

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