A published paper or arXiv preprint answers the main question of Erdős Problem #1207 (theorem erdos_1207 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.

The problem, as stated in formal-conjectures (Apache 2.0):

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.

References formal-conjectures lists for this problem (a starting point, not evidence):

• [BMP05] Brass, Peter and Moser, William and Pach, János, Research problems in discrete geometry. (2005), xii+499.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
• [LPZ26] Lee, S., Pohoata, C., and Zhu, D. G., The Minkowski grid has robustly many repeated distances. arXiv:2607.05374 (2026).
• [PaTa02] Pach, János and Tardos, Gábor, Isosceles triangles determined by a planar point set. Graphs Combin. (2002), 769--779.