OPEN
This is open, and cannot be resolved with a finite computation.
- $500
Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or even $\gg n/\sqrt{\log n}$?
The pinned distance problem, a stronger form of
[89]. The example of an integer grid show that $n/\sqrt{\log n}$ would be best possible.
It may be true that there are $\gg n$ many such points, or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in
[Er75f] Erdős conjectured\[\sum_{x\in A}d(x) \gg \frac{n^2}{\sqrt{\log n}},\]where $A\subset \mathbb{R}^2$ is any set of $n$ points.
In
[Er97e] Erdős offers \$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\gg n$ many such points.
In
[Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see
[89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.
The best known bound is\[\gg n^{c-o(1)},\]due to Katz and Tardos
[KaTa04], where\[c=\frac{48-14e}{55-16e}=0.864137\cdots.\]
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This page was last edited 15 October 2025.
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #604, https://www.erdosproblems.com/604, accessed 2026-01-16
