Erdős Problem #90

declared status 'disproved'. Formalized: yes. 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? Current best: In [Er82e] he offers \$300 for the upper bound$n^{1+o(1)}$. This would be the best possible, as is shown by a set of lattice points. The best known upper bound is$O(n^{4/3}), due to Spencer, Szemer\'{e}di, and Trotter \cite{SST84}. In \cite{Er83c} and \cite{Er85} Erd\H{o}s offers \250 for an upper bound of the form $n^{1+o(1)}$. Prize: $500. OEIS: A186705. Tags: distances, geometry.

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