160.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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 160

Reference: erdosproblems.com/160

Imports (1)

FormalConjecturesUtil

def Erdos160.erdos_160.h Documented declaration: h line 34 source
Source documentation
Let $h(n)$ be the smallest $k$ such that $\{1,\ldots,n\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Subtype, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, Fin, Set, And, LE.le, Set.IsAPOfLength, ENat, Set.ncard, Set.image
15 instances, coercions and auxiliary constants left out.
```
noncomputable def erdos_160.h (n : ℕ) : ℕ :=
…
```
theorem Erdos160.erdos_160.known_upper On Mathoverflow user leechlattice shows that $h(n) \ll n^{\frac 2 3}$ sorry line 48 source
Source documentation
On Mathoverflow user leechlattice shows that $h(n) \ll n^{\frac 2 3}$.

L4DEL explanation
On Mathoverflow user leechlattice shows that $h(n) \ll n^{\frac 2 3}$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos160.erdos_160.h, HPow.hPow, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_160.known_upper :
    (fun n => (erdos_160.h n : ℝ)) =O[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3) := by
…
```
Mentions: Erdos160.erdos_160.h
theorem Erdos160.erdos_160.better_upper Estimate $h(n)$ by finding a better upper bound sorry line 58 source
Source documentation
Estimate $h(n)$ by finding a better upper bound.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsBigO, Filter.atTop, Erdos160.erdos_160.h, Asymptotics.IsLittleO, HPow.hPow, HDiv.hDiv, OfNat.ofNat
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_160.better_upper :
    let upper_bound : ℕ → ℝ := answer(sorry)
…
```
theorem Erdos160.erdos_160.better_lower Estimate $h(n)$ by finding a better lower bound sorry line 68 source
Source documentation
Estimate $h(n)$ by finding a better lower bound.

depends on sorry what this means
Statement uses: Nat, Real, sorryAx, Bool.false, And, Asymptotics.IsBigO, Filter.atTop, Erdos160.erdos_160.h, GT.gt, OfNat.ofNat, Real.exp, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv, Asymptotics.IsLittleO
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_160.better_lower:
    let lower_bound : ℕ → ℝ := answer(sorry)
…
```
theorem Erdos160.erdos_160.variants.known_lower The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka KeMe23 imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$ sorry line 83 source
Source documentation
The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka KeMe23 imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$.

L4DEL explanation
The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka KeMe23 imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Real.exp, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv, Erdos160.erdos_160.h
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_160.variants.known_lower :
    ∃ c > 0, (fun (n : ℕ) => exp (c * log (n : ℝ) ^ ((1 : ℝ) / 12)))
    =O[atTop] fun n => (erdos_160.h n : ℝ):= by
…
```
Mentions: c, Erdos160.erdos_160.h