OPENNOTORIOUSgeometrycombinatorial geometry
Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(1/\log\log n)}$ many pairs which are distance 1 apart?
Notes: Best upper bound is O(n^{4/3}) by Spencer, Szemeredi, and Trotter. Valtr showed a non-Euclidean metric achieving n^{4/3} pairs, suggesting Euclidean-specific methods are needed.
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