Erdős Problem #503
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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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.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/503.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA175769
Original pageerdosproblems.com/503
Attempts
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Verified \(f(22)=276\), an explicit \(278\)-point construction in \(\mathbb R^{23}\), the exact \(d\leq8\) table, and the revised reduction \(f(d)=\max\{g(d),s(d)+1,s(d-1)+3\}\); the first remaining c
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