# P3022: Non-uniqueness of point sets minimizing distinct distances - ID: `P3022` - Reference: `erdos-problem-91` - Page: https://theoremdb.org/statements/P3022 - Record maturity: Reviewed problem with recorded work ## Problem Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. ### Context This is Erdős's Problem 91 from his 1987 paper on combinatorial and metric problems in geometry. Erdős conjectured that for large $n$ there exist at least two (and likely many) non-similar optimal configurations. The cases $n=3$ and $n=5$ are known to have unique minimizers (the equilateral triangle and regular pentagon, respectively), while non-uniqueness has been established for $n=4,6,7,8,9$. ### Problem setup - **Definition (For a finite set $A\subset\mathbb{R}^2$, the number of distinct distances).** For a finite set $A\subset\mathbb{R}^2$, the number of distinct distances is $d(A)=|\{\|x-y\|:x,y\in A, x\neq y\}|$. - **Definition (For each positive integer $n$, $m(n)=\min\{d(A):A\subset\mathbb{R}^2, |A|=n\}$).** For each positive integer $n$, $m(n)=\min\{d(A):A\subset\mathbb{R}^2, |A|=n\}$. - **Definition (A set $A$).** A set $A$ is optimal for $n$ if $|A|=n$ and $d(A)=m(n)$. - **Definition (Two sets $A,B\subset\mathbb{R}^2$ are similar if there exists a map $f:\mathbb{R}^2\to\mathbb{R}^2$ of the form $f(x)=r\cdot R(x)+t$ where $r>0$, $R$).** Two sets $A,B\subset\mathbb{R}^2$ are similar if there exists a map $f:\mathbb{R}^2\to\mathbb{R}^2$ of the form $f(x)=r\cdot R(x)+t$ where $r>0$, $R$ is an orthogonal transformation, and $t\in\mathbb{R}^2$, such that $f(A)=B$. - **Definition (A positive integer $n$ has a unique minimizer if every two optimal sets for $n$ are similar).** A positive integer $n$ has a unique minimizer if every two optimal sets for $n$ are similar. - **Remark.** This is Erdős's Problem 91 from his 1987 paper on combinatorial and metric problems in geometry. Erdős conjectured that for large $n$ there exist at least two (and likely many) non-similar optimal configurations. The cases $n=3$ and $n=5$ are known to have unique minimizers (the equilateral triangle and regular pentagon, respectively), while non-uniqueness has been established for $n=4,6,7,8,9$. ### What counts as a solution - Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. ## Status OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. [1](#reference-1) ## Work ### Evidence for the current status **Claim 1 (Current status and unresolved remainder).** OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. ### Background and intake notes - Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. - The maintained database entry for Erdős Problem 91 was open at commit 8138974387d9030542daabe67faaa33eff9356f8. - The pinned Formal Conjectures declaration was matched by problem number and source locator. - The controlled TheoremDB source corpus was checked for an already published record with the same slug. ### Open directions - **Route 1** (reported): Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. [1](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-91`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Erdős Problems database, Problem 91, maintained status record. Erdős Problems record 91, checked 2026-08-01. Problem 91; status field and linked bibliography https://www.erdosproblems.com/91 - Also cited at Problem 91; status snapshot 8138974387d9030542daabe67faaa33eff9356f8 - Also cited at See dataset.references[0] for the exact external source and locator. - Also cited at Editorial research route recorded 2026-07-31 - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source use: original_summary - Records the current open status and links the literature attached to this exact numbered problem. - Source used to formulate or check the problem record. - Source used to assess the problem's recorded status. - For Non-uniqueness of point sets minimizing distinct distances: Records the current open status and links the literature attached to this exact numbered problem. - Source named by the research packet. 2. Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 91 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source use: original_summary - Pins the maintained database snapshot used for this release's dated status decision. - Source used to assess the problem's recorded status. - For Non-uniqueness of point sets minimizing distinct distances: Pins the maintained database snapshot used for this release's dated status decision. 3. Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 91. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/91.lean:L147; theorem erdos_91; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/91.lean#L147 - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31 - Source use: original_summary - Supplies the pinned formal declaration whose human-readable editorial statement is published here. - Source used to assess the problem's recorded status. - For Non-uniqueness of point sets minimizing distinct distances: Supplies the pinned formal declaration whose human-readable editorial statement is published here.