Erdős problem 91 · problems.science

Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$ . Prove that for large $n$ there are at least two (and probably many) such $A$ which are non-similar.

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91.lean

Retained formal statement9 of 14

For $n = 5$ the regular pentagon is the unique such set (which has two distinct distances). Erdős mysteriously remarks in [Er90] this was proved by 'a colleague'. (In [Er87b] this is described as 'a colleague from Zagreb (unfortunately I do not have his letter)'.) A published proof of this fact is provided by Kovács [Ko24c].

FormalConjectures/ErdosProblems/91.leanErdos91.erdos_91.variants.five1 lineExact file

1Erdos91.UniqueMinimizer 5

SolvedStatement only, no proof

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