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ShowingImportancePrizesTopicErdős unit distance conjecture
Erdős's conjecture that the maximum number of unit-distance pairs among n points in the plane is n^(1+o(1)), i.e. grows sub-polynomially beyond linear. Claims about its truth or refutation, upper and lower bounds, candidate configurations or counterexamples, and the conjecture's lattice-based origins belong here.
OpenAI's disproof of the Erdős unit distance conjecture was generated in a single model run without human mathematical intervention
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryDiscrete geometryAI-assisted Erdős problem solvingimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
OpenAI's AI-generated disproof of the unit distance conjecture meets a top mathematics journal's acceptance standard
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →AI mathematical discoveryOpenAIAI-assisted Erdős problem solvingimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The unit distance counterexample was the first historically significant mathematical proof produced by an AI model.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →AI mathematical discoveryDiscrete geometryimportance · notableImportance 0.60, from 0 to 1 · notable: a contested point in a live debate (also the default before judging). Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
OpenAI's unit distance counterexample is a natural generalization of Erdős's lattice construction, introducing no new geometric tools
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Discrete geometryAI-assisted Erdős problem solvingAI mathematical discoveryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The published OpenAI unit distance manuscript is a human-edited exposition of the model's raw output, with references and explanatory material added afterward
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →OpenAIAI discovery creditAI mathematical discoveryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
OpenAI's unit distance paper adequately cites closely related prior ideas in the literature
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →OpenAIAI-assisted Erdős problem solvingAI discovery creditimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Erdős's unit distance problem was one of the most important long-standing open problems in discrete geometry
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Discrete geometryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
OpenAI's counterexample to the Erdős unit distance conjecture is mathematically correct
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryOpenAIDiscrete geometryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
ContradictedAvailable evidence weighs against the claim.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryimportance · minorImportance 0.42, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The maximum number of unit distances among n planar points is at least n^(1+c/log log n) for some constant c > 0.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Erdős's unit distance problem is equivalent to finding the maximum number of edges of a unit-distance graph on n vertices.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryGraph theoryimportance · settledImportance 0.10, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The maximum number of unit distances among n planar points is O(n^(4/3)).
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryimportance · settledImportance 0.22, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The exact asymptotic order of the maximum number of unit distances among n planar points is unknown.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The square grid is not asymptotically optimal for maximizing unit distances among n planar points.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The n-point square grid, however scaled, has at most n^(1+O(1/log log n)) unit distances.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Discrete geometryimportance · settledImportance 0.10, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Discrete geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
OpenAI researchers first examined the model's unit distance proof only after an automated grading pipeline had rated it correct
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryOpenAIAI-assisted Erdős problem solvingimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The prompt OpenAI gave its model for the unit distance problem contained no hints toward a counterexample or number-field methods.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryAI-assisted Erdős problem solvingimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
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