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Number of points in space bounded from above

Source: Romanian Master in Mathematics 2009, Problem 2

March 7, 2009
analytic geometrymodular arithmeticcombinatorics unsolvedcombinatoricsnumber theorycombinatorial geometry

Problem Statement

A set S S of points in space satisfies the property that all pairwise distances between points in S S are distinct. Given that all points in S S have integer coordinates (x,y,z) (x,y,z) where 1≤x,y,z≤n, 1 \leq x,y, z \leq n, show that the number of points in S S is less than \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).
Dan Schwarz, Romania
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