Problem 1207

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 $c>0$ ?

Source: erdosproblems.com/1207

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open. The site's label is OPEN (page last edited 7 September 2026). Its remark of that date credits [LPZ26] with answering the particular question and keeps the problem open for the general estimate of $P_d(n)$ .

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: S. Lee, C. Pohoata, and D. G. Zhu, 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. 18 (2002), 769-779.

Problem 1207

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