Pairs of points exactly 1 unit apart in the plane

Question

I don't have an idea how to prove, that between any n points on the plane, there are not more than $O(n^\frac{3}{2})$ pairs with distance of 1 unit between each other... Thanks a lot for any help!

Answer 1

Weaker result :

Take a graph whose vertices are your $n$ points. We say that there is an edge joining two vertices iff the corresponding points are at distance $1$.

Then it is easy to prove that in the plane there cannot be $4$ points such that the distance between every pair of them is $1$.

Hence your graph is $K_4$ free, and you can use Turan's theorem (see Turan Theorem) : the number of edges is at most $\frac{1}{3}n^2$

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After some researchs, the references $[48]$ and $[49]$ in this article (Distinct distances in graph drawings) should be useful : the bound they proved is proportional to $n^{\frac{4}{3}}$.