Erdős's unit distance problem is equivalent to finding the maximum number of edges of a unit-distance graph on n vertices.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The Erdős unit distance problem asks for u(n), the maximum number of pairs at the same distance among n points in the plane. A unit-distance graph is a graph whose vertices can be placed at distinct points of the plane so that adjacent vertices are exactly one unit apart, so the number of edges in a unit-distance graph on n vertices is exactly a count of unit-distance pairs among n points, and the densest such graph has u(n) edges. The two formulations therefore differ only in vocabulary, and the equivalence holds by unpacking the definitions rather than by any substantive theorem.
Two small points make the equivalence robust rather than merely conventional. Because the problem is invariant under scaling, "the same distance" can be taken to be one without loss of generality. And it does not matter whether a unit-distance graph is required to contain every unit-distance pair of its point set as an edge or is allowed to omit some: the maximum edge count is the same under both conventions, since the densest graph on a given point set is the one that includes every unit pair. The research literature routinely adopts the graph formulation as the definition of u(n) itself, identifying it with sequence A186705 in the On-Line Encyclopedia of Integer Sequences, and no source disputes the identification.
Full reasoning: the evidence and decisions behind this verdict
The claim is an equivalence between two formulations of one extremal problem, and it is settled by definitions.
Formulation one: u(n) is the maximum, over sets P of n distinct points in the plane and over distances d > 0, of the number of pairs in P at distance exactly d. Since a dilation by 1/d carries pairs at distance d to pairs at distance 1 and preserves the number of points, the maximum is attained with d = 1, so u(n) is the maximum number of unit-distance pairs among n planar points.
Formulation two: a unit-distance graph on n vertices is a graph admitting an injection of its vertex set into the plane under which every edge joins points at distance 1. The edge set of any such graph is a subset of the unit-distance pairs of its n-point image, so it has at most u(n) edges; conversely, the graph on an extremal point set with all unit pairs as edges is a unit-distance graph with u(n) edges. Hence the maximum edge count of a unit-distance graph on n vertices equals u(n). The same argument shows the maximum is unchanged if one insists on "faithful" (strict) unit-distance graphs, where every unit pair must be an edge; the paper "Probabilistic formulation of the Hadwiger–Nelson problem" (arxiv.org/pdf/2112.07665) states and proves exactly this as its Proposition 1.2.2.
Instances: the MathWorld entry (mathworld.wolfram.com/ErdosUnitDistanceProblem.html) asserts the equivalence directly in its opening lines, without argument, as expected for a definitional reformulation. Alexeev, Mixon and Parshall (arxiv.org/abs/2412.11914) open their treatment of the problem by defining u(n) as the maximum edge count of a unit-distance graph on n vertices and identifying it with OEIS A186705, the same sequence MathWorld gives for u(n); this is the equivalence adopted as a working definition. No source found denies or qualifies the identification.
The verdict does not depend on the current state of the problem itself (the 2026 disproof of the conjectured upper bound, or the still-unknown asymptotic order of u(n)); those bear on the value of u(n), not on whether the two formulations define the same quantity. What would change the conclusion: only a demonstration that some standard usage of "unit-distance graph" (for example one permitting coincident points or a non-Euclidean metric) yields a different extremal function, and no such usage appears in the literature on this problem.
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Provenance
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It is equivalent to finding a maximally dense unit-distance graph on n vertices.
Reformulation of the unit distance problem.
Asserted without evidence of the source's own. The encyclopedia entry states the reformulation as a matter of definition and offers no argument for it; none is needed, since the two formulations differ only in vocabulary.
Let U(n) denote the set of unit-distance graphs on n vertices, and let u(n) := max{|E(G)| : G ∈ U(n)} denote the maximum number of edges in such a graph. (This is known as A186705(n) in the On-Line Encyclopedia of Integer Sequences [12].)
The paper introduces the Erdős unit distance problem and immediately defines its extremal function u(n) as the maximum edge count of a unit-distance graph on n vertices, identifying it with OEIS A186705; the equivalence is asserted by way of definition rather than stated as a theorem.
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.