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1000 points with distinct pairwise distances

Source: Iran 3rd round 2012-Special Lesson exam-Part1-P3

July 27, 2012
inductionpigeonhole principlegeometryperpendicular bisectoralgebrabinomial theoremcombinatorics proposed

Problem Statement

Prove that if nn is large enough, among any nn points of plane we can find 10001000 points such that these 10001000 points have pairwise distinct distances. Can you prove the assertion for nαn^{\alpha} where α\alpha is a positive real number instead of 10001000?
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