748.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 748

References:

erdosproblems.com/748
[CaEr90] Cameron, P. J. and Erdős, P., On the number of sets of integers with various properties. (1990), 61-79.
[Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.
[Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.
[Gr04] Green, Ben, The Cameron-Erdős conjecture. Bull. London Math. Soc. (2004), 769-778.
[Sa03] Sapozhenko, A. A., The Cameron-Erdős conjecture. Dokl. Akad. Nauk (2003), 749-752.

Imports (1)

FormalConjecturesUtil

def Erdos748.f f n counts the number of sum-free $A\subseteq \{1,\ldots,n\}$, i.e. $A$ contains no solutions to $a=b+c$ with $a,b,c\in A$ line 44 source
Source documentation
f n counts the number of sum-free $A\subseteq \{1,\ldots,n\}$, i.e. $A$ contains no solutions to $a=b+c$ with $a,b,c\in A$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset, Finset.filter, IsSumFree, Classical.propDecidable, Finset.powerset, Finset.Icc, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos748.erdos_748 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Real.logb, OfNat.ofNat, Erdos748.f, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos748.erdos_748.variants.lower_bound It is trivial to see that $f(n) \geq 2^{\frac{n}{2}}$, considering all subsets of $(n/2,n]$ sorry line 67 source
Source documentation
It is trivial to see that $f(n) \geq 2^{\frac{n}{2}}$, considering all subsets of $(n/2,n]$. The source writes $[n/2,n]$, but for even $n$ that interval is not sum-free, since $n/2 + n/2 = n$.

depends on sorry what this means
Statement uses: Nat, LE.le, Real, HPow.hPow, OfNat.ofNat, HDiv.hDiv, Erdos748.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_748.variants.lower_bound (n : ℕ) :
    (2 : ℝ) ^ ((n : ℝ) / 2) ≤ (f n : ℝ) := by
…
```
Mentions: Erdos748.f
theorem Erdos748.erdos_748.variants.green_sapozhenko Green [Gr04] and Sapozhenko [Sa03] proved that $f(n) \ll 2^{n/2}$ sorry line 73 source
Source documentation
Green [Gr04] and Sapozhenko [Sa03] proved that $f(n) \ll 2^{n/2}$.

depends on sorry what this means
Statement uses: Exists, Real, Nat, LE.le, Erdos748.f, HMul.hMul, HPow.hPow, OfNat.ofNat, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_748.variants.green_sapozhenko :
    ∃ C : ℝ, ∀ n : ℕ, (f n : ℝ) ≤ C * 2 ^ ((n : ℝ) / 2) := by
…
```
Mentions: Erdos748.f
theorem Erdos748.erdos_748.variants.asymptotic In fact, both Green [Gr04] and Sapozhenko [Sa03] prove the stronger asymptotic $f(n) \sim c_n 2^{n/2}$, where $c_n$ takes on one of two values depending on the parity of $n$ sorry line 81 source
Source documentation
In fact, both Green [Gr04] and Sapozhenko [Sa03] prove the stronger asymptotic $f(n) \sim c_n 2^{n/2}$, where $c_n$ takes on one of two values depending on the parity of $n$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Tendsto, Nat, HDiv.hDiv, Erdos748.f, HMul.hMul, HPow.hPow, Filter.atTop, nhds, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_748.variants.asymptotic :
    ∃ c₀ c₁ : ℝ, 0 < c₀ ∧ 0 < c₁ ∧
      Tendsto (fun m : ℕ ↦ (f (2 * m) : ℝ) / 2 ^ m) atTop (nhds c₀) ∧
      Tendsto (fun m : ℕ ↦ (f (2 * m + 1) : ℝ) / 2 ^ ((2 * m + 1 : ℝ) / 2)) atTop
        (nhds c₁) := by
…
```
Mentions: Erdos748.f