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ShowingImportancePrizesTopicDiscrete geometry
The study of finite point configurations and their combinatorial and metric properties: unit and distinct distances, incidence and convexity problems, lattice versus general point constructions, and bounds on repeated geometric configurations. Claims about distance counts, lower or upper bound constructions, and point-set extremal problems belong here; purely algebraic or topological geometry clai
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 discoveryErdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureAI-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 →
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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureAI mathematical discoveryOpenAIimportance · 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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureGraph 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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureimportance · 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 →Erdős unit distance conjectureimportance · 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'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 →Erdős unit distance conjectureAI mathematical discoveryAI-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 →
There exist n-point planar sets where every subset has a distance repeated at least c|A|^2/n^(1-delta) times.
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 →Repeated distances problemimportance · 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 packing approach to proving Erdős–Szekeres-type results originates in Seidenberg's 1959 paper.
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 →CombinatoricsHappy ending problemimportance · 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 →
For k≥1 and integer c with k^2+2c+1≥1, f(k^2+2c+1) ≤ k + c/k.
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 →Square packing constantimportance · 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 →
For all k≥1, the axis-parallel square-packing constant satisfies f(k^2+1) ≤ k.
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 →Square packing constantimportance · 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 →
For all n, the coin-game constant satisfies c(n) ≥ 1/f(n), where f is the square-packing constant.
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 →Coin-game constantSquare packing constantimportance · 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 →
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