Problem 217

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claims/: The 4 claim pages of Problem 217, one per claimant's result; the problem's standing derives from them.

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Statement. For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?

Status. Open, the site's label (OPEN). Four claim pages are recorded, each answering one value of $n$ yes: $n=5$ on [[problems/distance_problems/E0217/claims/1983_01_01_pomerance|Pomerance's claim page]], a construction Erdős describes in [Er83c], which stays claimed because that transcript is its only publication; $n=7$ on [[problems/distance_problems/E0217/claims/1987_01_01_palasti|Palásti's 1987 claim page]]; $n=6$, with the further restrictions that no point is equidistant from three others and no triangle is equilateral, on [[problems/distance_problems/E0217/claims/1989_01_01_palasti|Palásti's Discrete Mathematics claim page]]; and $n=8$ on [[problems/distance_problems/E0217/claims/1989_09_01_palasti|Palásti's lattice-point claim page]], the last three accepted on their refereed publications. With the four-point example, an isosceles triangle and its circumcenter ([Er83c], p. 53; Erdős's own example, recorded in prose rather than on a claim page), the answer is yes for $4\le n\le8$ and open for every $n\ge9$. The property does not pass to subsets, so each construction settles only its own $n$. Erdős believed that no example exists for all large $n$; this would follow from $h(n)\ge n$ for large $n$, with $h$ the function of Problem 98.

Source. erdosproblems.com/217, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #217, https://www.erdosproblems.com/217.

References.

• [Er83c] Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.
• [Pa87] Palásti, Ilona, On the seven points problem of P. Erdős. Studia Sci. Math. Hungar. (1987), 447-448.
• [Pa89] Palásti, Ilona, A distance problem of P. Erdős with some further restrictions. Discrete Math. (1989), 155-156.
• [Pa89b] Palásti, I., Lattice-point examples for a question of Erdős. Period. Math. Hungar. (1989), 231-235.

Formalization. None recorded.

Current assessment

The instances $n=4,5,6,7,8$ are answered yes by the constructions on the claim pages and the four-point example; every $n\ge9$ is open. [Er83c] records that Erdős had believed no example with more than four points exists, that Pomerance's five points corrected him, and that a Hungarian high-school student had found an unpublished six-point example before Palásti's published one. The only nine-point configuration recorded here is the relaxed witness below, which has the required distance multiplicities but one collinear triple and two concyclic quadruples, so it does not answer $n=9$. Burt, Goldstein, Manski, Miller, Palsson and Suh, Crescent configurations (arXiv:1509.07220; card), record in Remark 3.1 that an exhaustive search of a 91-point hexagonal region of the triangular lattice found no nine-point example, and prove that $n$ points with the required multiplicities exist in $\mathbb R^{n-2}$ for every $n\ge3$, which says nothing about the plane.

Progress

The nine-point configuration below is a relaxed near-miss: it has the required distance multiplicities but fails general position, so it does not answer the question for $n=9$. It was posted by IheartTesla in comment 8735, dated 4 September 2026. The author reports that an AI agent found it while minimizing the number of collinear triples and concyclic quadruples. The account below checks the supplied witness, not that search or any optimality claim.

Exact witness and distance multiplicities

Interpret a lattice pair $(a,b)$ as the Euclidean point $(a+b/2,\sqrt{3}b/2)$. In the author's numbering the pairs are

$$\begin{aligned} P_0&=(4,-2),&P_1&=(4,-3),&P_2&=(2,1),\\ P_3&=(3,-1),&P_4&=(2,2),&P_5&=(-1,0),\\ P_6&=(0,0),&P_7&=(1,-1),&P_8&=(3,1). \end{aligned}$$

For a difference $(a,b)$, the squared distance is $q(a,b)=a^2+ab+b^2$. The following table lists all 36 unordered pairs; $ij$ means $\{P_i,P_j\}$.

Thus there are eight distinct distances, with multiplicities a permutation of $1,\ldots,8$. The required ordering is by multiplicity, not by length.

General-position failures

For lattice coordinates, collinearity is tested by the determinant with rows $(1,a,b)$. Concyclicity of four points, provided some three are noncollinear, is tested by the determinant with rows $(1,a,b,q(a,b))$. These are exact integer tests: a Euclidean circle has an equation $q(a,b)+u a+v b+w=0$ in this basis.

Direct calculation over the 84 triples and 126 quadruples gives precisely these zero determinants:

Every triple within either listed quadruple is noncollinear, so neither circle determinant vanishes merely because four points lie on a line. Consequently the witness has exactly one collinear triple and two concyclic quadruples, or $(C,Q)=(1,2)$. These are three failed general-position tuples; they are not three extra distance values.

Scope and remaining leads

The exact check above concerns one supplied finite witness and is the corpus's own computation, not an outside review. No lower bound on $C+Q$, general-position construction or nonexistence theorem follows from it. Equality of some distance values with Palásti's eight-point example does not identify this point set as an extension of that construction.

The same post's reports of unsuccessful searches for $C+Q\le2$, for a ten-point example and for off-lattice examples are search reports without a published record. Sallerk's comment 8674 points to the paper of Burt, Goldstein, Manski, Miller, Palsson and Suh cited in the Current assessment and reports a larger triangular-lattice exclusion search, with AI assistance, through squared diameter 400; that search has no published record either, and the paper's own Remark 3.1 covers a 91-point hexagonal region.

Known Results

• $n=4$: an isosceles triangle with its circumcenter, three distances with multiplicities $3$, $2$ and $1$ ([Er83c], p. 53).
• $n=5$: Pomerance's construction, a unit equilateral triangle, its circumcenter and a point on the unit circle about one vertex equidistant from the circumcenter and another vertex, multiplicities $4,3,2,1$ ([Er83c], p. 54; claim page).
• $n=6$: Palásti [Pa89], with no point equidistant from three others and no equilateral triangle (claim page); [Er83c] reports an earlier unpublished six-point example.
• $n=7$: Palásti [Pa87] (claim page).
• $n=8$: Palásti [Pa89b], on the triangular lattice, with lattice examples for every $n\le7$ (claim page).
• $n\ge9$: open. The relaxed nine-point witness above fails general position and is not a positive $n=9$ case.

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.

• bhowmick_2024_problem_erdos_about_rich_distances
• burt_2015_crescent_configurations
• burt_2015_crescent_configurations / definition_1_2
• burt_2015_crescent_configurations / remark_3_1
• burt_2015_crescent_configurations / theorem_1_3
• erdos_1983_combinatorial_problems_geometry
• erdos_1983_combinatorial_problems_geometry / question_p53