Problem 657

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claims/: The 2 claim pages of Problem 657, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that if $A\subset \mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\to \infty$?

Status. Open. The site labels the problem OPEN (page last edited 15 October 2025). The assertion is proved for collinear sets, which are the progression-free sets of reals: [[problems/distance_problems/E0657/claims/2008_12_01_dumitrescu|Dumitrescu's accepted partial claim]] gives $(\log n)^c\le f(n)\le2^{O(\sqrt{\log n})}$ on that class, and [[problems/distance_problems/E0657/claims/2025_08_19_hunter|Hunter's pending partial claim]] raises the lower bound to $2^{c(\log n)^{1/9}}$. No superlinear lower bound is known for planar sets in general, so the standing derived from the claim pages is open.

Source. erdosproblems.com/657, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #657, https://www.erdosproblems.com/657.

References.

• [BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).
• [Du08] Dumitrescu, Adrian, On distinct distances and $\lambda$-free point sets. Discrete Math. (2008), 6533-6538.
• [Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
• [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
• [KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).

Formalization. None recorded.

Current assessment

A literature check on 6 September 2026 covered the local-distance, progression-free, and lattice-box sources discussed below and searched for later resolutions of the no-isosceles planar question. No exact-target resolution was located; this bounded search does not itself prove openness.

The site's commentary also records Straus's observation that for $2^k\ge n$ there are $n$ points in $\mathbb R^k$ with no isosceles triangle and at most $n-1$ distances, so the question is particular to the plane and low dimensions.

Progress

The planar question asks for a superlinear number of distances. The immediate lower bound is $n-1$: from any fixed point, the distances to the remaining points are all distinct. Write $D(n,3,3)$ for the smallest distance count among sets satisfying the three-point condition. Fox, Pach, and Suk record $D(n,3,3)\geq n-1$ and constructions with $D(n,3,3)\leq n\exp(O(\sqrt{\log n}))$ in their introduction (pp. 1--2). See Fox, Pach, and Suk's paper (card). Their stronger local-distance theorem concerns other parameters and does not settle this three-point condition.

For $n\geq3$, Quanyu Tang's 2 October 2025 discussion comment records the parity refinement $D(n,3,3)\geq n$ when $n$ is odd, retaining $n-1$ when $n$ is even. Its counting argument observes that every distance class is a matching, with at most $\lfloor n/2\rfloor$ edges. This elementary refinement still has linear order and does not settle the asymptotic question.

Known Results

On a line, the condition is exactly the absence of a nontrivial three-term arithmetic progression. The distance count is then $(|A-A|-1)/2$, and collinear sets are admissible planar sets, so the one-dimensional results are special cases of the problem. In [Er73] Erdős said that $f(n)\to\infty$ was not known even on the line; that case is now settled. Dumitrescu's paper proves $(\log n)^c\le f(n)\le2^{O(\sqrt{\log n})}$ for collinear sets (claim page), and the thread's deduction from Ruzsa's result and the progression-free bounds of Kelley–Meka and Bloom–Sisask raises the lower bound to $2^{c(\log n)^{1/9}}$ (claim page). Dumitrescu's upper bound also bounds the planar quantity. The later preprint Raghavan (arXiv:2603.27045v3, 15 May 2026, abstract) states the progression-free bound $|A|\leq N\exp(-c(\log N)^{1/6}/\log\log N)$ for progression-free $A\subseteq[N]$ as $N\to\infty$, with an absolute constant $c>0$. These are statements about one-dimensional additive structure; none gives a superlinear lower bound for planar sets off a line.

The October 2025 discussion contains an attempted planar application of these estimates, followed by Quanyu Tang's objection and Alfaiz's acknowledgment that the improvement applies to the one-dimensional or torsion-free-group problem. The later Raghavan-based comment of 6 August 2026 has the same scope and is disclosed on Hunter's claim page.

There is also a distinct lattice-box problem. Croot, Mao, Pohoata, Sheffer, and Yip's combinatorial large-sieve paper (arXiv:2606.17487v2), dated 24 June 2026, bounds the size of a subset of $[N]^2$ containing no isosceles triangle by

$$N^2\exp\!\left(-c\frac{\log N}{\log\log N}\right)$$

up to an absolute constant (Theorem 1.4). Their convention includes equally spaced collinear triples. The method uses the fact that each fixed-distance graph is a matching, together with a combinatorial large sieve. A bound on the size of a subset of a fixed arithmetic box does not establish a superlinear distance lower bound for an arbitrary real planar set.

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
• bloom_2023_improvement_kelley_meka_bounds_three_term
• kelley_2023_strong_bounds_3_progressions
• dumitrescu_2008_distinct_distances_points_general_position
• fox_2016_more_distinct_distances_local_conditions
• fox_2016_more_distinct_distances_local_conditions / d_n_3_3_bounds_pp1_2
• fox_2016_more_distinct_distances_local_conditions / theorem_1
• sheffer_2014_distinct_distances_open_problems_current_bounds
• sheffer_2014_distinct_distances_open_problems_current_bounds / problem_28
• erdos_1979_problems_results_graph_theory_combinatorial_analysis
• erdos_1979_problems_results_graph_theory_combinatorial_analysis / question_p158