Erdős's unit distance problem is equivalent to finding the maximum number of edges of a unit-distance graph on n vertices.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Left the claim atomic: the equivalence rests only on the definition of a unit-distance graph and on scale invariance, both stipulative setup rather than reusable propositions of the discourse, so no subclaims pass the claim bar. No upward proposition to mint: the claim is a reformulation, not an argument for or meta-claim about a theorem; recorded a see-also to the claim that the asymptotic order of u(n) is unknown so readers reach the problem's live status. Read the MathWorld source whole and recorded its reading (asserts without evidence, as a definitional statement should); added a second instance from Alexeev–Mixon–Parshall (arXiv 2412.11914), which adopts the equivalence as its definition of u(n); wrote an immaterial source map. Assessed verified at 0.97 confidence, credence 0.99, marginal yield 0.02. Importance set to 0.10 with contestation 0.02: settled scaffolding inside a live problem. Canonical form sharpened to name Erdős and state the quantity as maximum edge count. No dependents exist, so no notification. No formal statement drafted: the claim is a definitional equivalence between an informal problem and its graph formulation, and formalizing it would amount to restating one definition, with no value for attempts or prizes.
Assessed Verified
verdict confidence 0.97 · credence 0.99
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.
Claim entered the graph