166.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 166

References:

erdosproblems.com/166
[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.
[AKS80] Ajtai, M., Komlós, J. and Szemerédi, E., A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.
[MaVe23] Mattheus, S. and Verstraëte, J., The asymptotics of $r(4,t)$. Ann. of Math. (2024), 941-965.

Imports (1)

FormalConjecturesUtil

theorem Erdos166.erdos_166 sorry line 52 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, GE.ge, SimpleGraph.classicalRamsey, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos166.erdos_166.variants.mattheus_verstraete Mattheus and Verstraëte [MaVe23] proved the explicit lower bound $$R(4,k) \gg \frac{k^3}{(\log k)^4}.$$ sorry line 64 source
Source documentation
Mattheus and Verstraëte [MaVe23] proved the explicit lower bound $$R(4,k) \gg \frac{k^3}{(\log k)^4}.$$

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, GE.ge, SimpleGraph.classicalRamsey, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_166.variants.mattheus_verstraete :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ (k : ℕ) in atTop,
      (SimpleGraph.classicalRamsey 4 k : ℝ) ≥ C * (k : ℝ) ^ 3 / (Real.log k) ^ 4 := by
…
```
Mentions: SimpleGraph.classicalRamsey
theorem Erdos166.erdos_166.variants.mattheus_verstraete_implies The bound of Mattheus and Verstraëte answers erdos_166, with logarithmic exponent $4$ line 71 source
Source documentation
The bound of Mattheus and Verstraëte answers erdos_166, with logarithmic exponent $4$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, GE.ge, SimpleGraph.classicalRamsey, HDiv.hDiv, HMul.hMul, HPow.hPow, Real.log, Filter.atTop, Iff, True
Proof uses: Iff.intro, Exists.intro, And.intro, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Bool.false, Filter.Eventually.mono, Eq.mpr, LE.le, id, Eq, Eq.trans, congrArg, Real.rpow_ofNat, Eq.mp, trivial
54 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_166.variants.mattheus_verstraete_implies :
    type_of% erdos_166.variants.mattheus_verstraete → type_of% erdos_166 := by
…
```
Mentions: Erdos166.erdos_166.variants.mattheus_verstraete, Erdos166.erdos_166