Claims
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KindShowingImportancePrizesTopicDiscrete 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 maximum number of unit distances among n planar points is at least n^(1+c/log log n) for some constant c > 0.
VerifiedThe evidence, examined directly, establishes 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 · 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 evidence, examined directly, establishes 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 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 evidence, examined directly, establishes 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 · 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 square grid is not asymptotically optimal for maximizing unit distances among n planar points.
VerifiedThe evidence, examined directly, establishes 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 0.30, 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 arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.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 maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
ContradictedThe evidence, examined directly, 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 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 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 →
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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