142.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 142

Reference: erdosproblems.com/142

Imports (1)

FormalConjecturesUtil

abbrev Erdos142.r line 32 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set.IsAPOfLengthFree.maxCard
theorem Erdos142.erdos_142 Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression sorry line 40 source
Source documentation
Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression. That is, find $f_k$ with $r_k(N) / f_k(N) \to 1$ as $N \to \infty$.

depends on sorry what this means
Statement uses: Nat, Asymptotics.IsEquivalent, Real, Filter.atTop, Erdos142.r, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_142 (k : ℕ) : (fun N => (r k N : ℝ)) ~[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos142.r
theorem Erdos142.erdos_142.variants.lower Documented declaration: lower sorry line 48 source
Source documentation
Show that $r_k(N) = o_k(N / \log N)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Asymptotics.IsLittleO, Real, Filter.atTop, Erdos142.r, HDiv.hDiv, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_142.variants.lower (k : ℕ) (hk : 1 < k) :
    (fun N => (r k N : ℝ)) =o[atTop] (fun N : ℕ => N / (N : ℝ).log) := by
…
```
Mentions: Erdos142.r
theorem Erdos142.erdos_142.variants.isTheta Determine the order of magnitude of $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression sorry line 59 source
Source documentation
Determine the order of magnitude of $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression. That is, find $f_k$ with $r_k(N) \asymp_k f_k(N)$.

depends on sorry what this means
Statement uses: Nat, Asymptotics.IsTheta, Real, Filter.atTop, Erdos142.r, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_142.variants.isTheta (k : ℕ) :
    (fun N => (r k N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos142.r
theorem Erdos142.erdos_142.variants.three Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression sorry line 72 source
Source documentation
Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression. That is, find $f$ with $r_3(N) / f(N) \to 1$ as $N \to \infty$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos142.r, OfNat.ofNat, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_142.variants.three : (fun N => (r 3 N : ℝ)) ~[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos142.r