Problem 1207

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claims/: The 1 claim page of Problem 1207, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $c>0$?

Status. 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 $P_d(n)$.

Source. erdosproblems.com/1207, accessed 2026-09-04 and 2026-10-07. Cite as: T. F. Bloom, Erdős Problem #1207, https://www.erdosproblems.com/1207.

References.

• [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.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation asks for estimates of $P_d(n)$, the size of an isosceles-free subset that every set of $n$ points in $\mathbb{R}^d$ is guaranteed to contain, and in particular whether $P_2(n)<n^{1-c}$ for some $c>0$. The site's label is OPEN.

Known bounds. By the site's remarks, Erdős [Er80] attributes the problem to Riddell and notes that $P_d(n)>n^{\epsilon_d}$ with $\epsilon_d\to 0$ as $d\to\infty$: a subset with all distances distinct contains no isosceles triangle, so $P_d(n)\ge f_d(n)\ge n^{1/(3d-3)-o(1)}$, with $f_d(n)$ as in Problem 1208. In the plane, the bound of [PaTa02], Theorem 1 on the number of isosceles triangles, combined with random deletion, gives $P_2(n)\gg_\varepsilon n^{2e/(5e-1)-\varepsilon}$ for every $\varepsilon>0$, where $2e/(5e-1)\approx0.4318$. The case $d=1$ asks for sets without three-term arithmetic progressions, the case $k=3$ of Problem 201.

Claims. One result is claimed from outside the project. Lee, Pohoata and Zhu's Minkowski grid construction (arXiv, 6 July 2026, found with the help of ChatGPT) gives a planar set of $n$ points in which every subset of at least $n^{1-\delta}$ points contains an isosceles triangle, so $P_2(n)<n^{1-c}$ for some $c>0$. It answers the particular question and leaves the general estimate open; it is a preprint without journal publication or Lean proof, so it stays claimed.

Not recorded as a claim. Section 5.3 of [BMP05] asserts that the regular polygon gives $P_2(n)\ll n^{1/2}$ but gives no details, and the site disputes the assertion: the isosceles triangles on the vertices of a regular polygon correspond to three-term progressions among the vertex indices, so the polygon bounds $P_2(n)$ only by about $r_3(n)$, the largest size of a progression-free subset of $\{1,\ldots,n\}$. The assertion has no proof to record.

Search scope: the site's problem page and the arXiv record and text of [LPZ26]. Not searched: zbMATH and MathSciNet.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• croot_2026_combinatorial_large_sieve_sidon_sets_distances
• croot_2026_combinatorial_large_sieve_sidon_sets_distances / theorem_1_4
• croot_2026_combinatorial_large_sieve_sidon_sets_distances / theorem_1_5
• pach_2002_isosceles_triangles_determined_planar_point_set
• pach_2002_isosceles_triangles_determined_planar_point_set / theorem_1
• pach_2002_isosceles_triangles_determined_planar_point_set / theorem_2
• sheffer_2014_distinct_distances_open_problems_current_bounds
• sheffer_2014_distinct_distances_open_problems_current_bounds / problem_26