Erdős problem 89 · problems.science

Erdős [Er46] asked whether every set of $n$ distinct points in $\mathbb{R}^2$ determines $\gg \frac{n}{\sqrt{\log n}}$ many distinct distances.

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

89.lean

Retained formal statement3 of 4

This theorem provides a sanity check, showing that the main conjecture (erdos_89) is strictly stronger than the solved Guth and Katz result. It proves that, trivially, if the lower bound $\frac{n}{\sqrt{\log n}}$ holds, then the weaker lower bound $\frac{n}{\log n}$ must also hold.

FormalConjectures/ErdosProblems/89.leanErdos89.erdos_89.variants.implies_n_dvd_log_n2 linesExact file

1((fun n => ↑n / √(Real.log ↑n)) =O[Filter.atTop] fun n => ↑(EuclideanGeometry.minimalDistinctDistances n)) →2  (fun n => ↑n / Real.log ↑n) =O[Filter.atTop] fun n => ↑(EuclideanGeometry.minimalDistinctDistances n)

TestStatement only, no proof

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