Erdős Problem #1207
OPENdatabase updated 2026-04-04
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`P d n` is the largest number \(m\) such that every set of \(n\) points in \(\mathbb{R}^d\) has an isosceles-free subset of size at least \(m\).
Let \(P_d(n)\) be such that in any set of \(n\) points in \(\mathbb{R}^d\) there exist at least \(P_d(n)\) many points which do not contain an isosceles triangle. Estimate \(P_d(n)\) - in particular, is it true that \[P_2(n)<n^{1-c}\] for some constant \(c>0\)? Lee, Pohoata, and Zhu [LPZ26] proved the stated planar bound. Their isosceles-free subsets also exclude degenerate triangles of equally spaced collinear points.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/1207.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1207
Attempts
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Lee--Pohoata--Zhu, arXiv:2607.05374v1, source-audited modulo the Hajir--Maire--Ramakrishna/Chebotarev tower input, proves \(P_2(n)<n^{1-c}\) for some \(c>0\) and all sufficiently large \(n\); the full
FOUND