Erdős problem 503 · problems.science

What is the size of the largest $A \subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x$ , $y$ , $z$ from $A$ , at least two of the distances $|x - y|$ , $|y - z|$ , $|x - z|$ are equal.

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5 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

503.lean

Retained formal statement2 of 5

Alweiss has observed a lower bound of $\binom{n + 1}{2}$ follows from considering the subset of $\mathbb{R}^{n + 1}$ formed of all vectors $e_i + e_j$ where $e_i$ , $e_j$ are distinct coordinate vectors. This set can be viewed as a subset of some $\mathbb{R}^n$ , and is easily checked to have the required property.

FormalConjectures/ErdosProblems/503.leanErdos503.erdos_503.variants.lower_bound1 lineExact file

1∀ (n : ℕ), (n + 1).choose 2 ≤ sSup {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x}

SolvedStatement only, no proof

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