877.lean`

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

6 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 877

References:

erdosproblems.com/877
[CaEr90] Cameron, P. J. and Erdős, P., On the number of sets of integers with various properties. (1990), 61-79.
[Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.
[LuSc01] Łuczak, Tomasz and Schoen, Tomasz, On the number of maximal sum-free sets. Proc. Amer. Math. Soc. (2001), 2205--2207.
[BLST15] Balogh, József and Liu, Hong and Sharifzadeh, Maryam and Treglown, Andrew, The number of maximal sum-free subsets of integers. Proc. Amer. Math. Soc. (2015), 4713--4721.
[BLST18] Balogh, József and Liu, Hong and Sharifzadeh, Maryam and Treglown, Andrew, Sharp bound on the number of maximal sum-free subsets of integers. J. Eur. Math. Soc. (JEMS) (2018), 1885--1911.

Imports (1)

FormalConjecturesUtil

def Erdos877.IsMaximalSumFree A is a maximal sum-free subset of {1, …, n} line 45 source
Source documentation
A is a maximal sum-free subset of {1, …, n}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: And, LE.le, Finset.Icc, OfNat.ofNat, IsSumFree, Eq
9 instances, coercions and auxiliary constants left out.
```
def IsMaximalSumFree (n : ℕ) (A : Finset ℕ) : Prop :=
…
```
def Erdos877.fm fm n counts the number of maximal sum-free subsets $A\subseteq\{1,\ldots,n\}$ line 51 source
Source documentation
fm n counts the number of maximal sum-free subsets $A\subseteq\{1,\ldots,n\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Finset, Finset.filter, Erdos877.IsMaximalSumFree, Classical.propDecidable, Finset.powerset, Finset.Icc, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
noncomputable def fm (n : ℕ) : ℕ :=
…
```
theorem Erdos877.erdos_877 sorry line 67 source
depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos877.fm, HPow.hPow, OfNat.ofNat, HDiv.hDiv
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 Erdos877.erdos_877.variants.luczak_schoen sorry line 75 source
depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, HDiv.hDiv, OfNat.ofNat, Nat, Filter.Eventually, LE.le, Erdos877.fm, HPow.hPow, HMul.hMul, 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 Erdos877.erdos_877.variants.cameron_erdos Cameron and Erdős [CaEr90] proved that $f_m(n)>2^{n/4}$ sorry line 81 source
Source documentation
Cameron and Erdős [CaEr90] proved that $f_m(n)>2^{n/4}$.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, LT.lt, Real, HPow.hPow, OfNat.ofNat, HDiv.hDiv, Erdos877.fm, Filter.atTop
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_877.variants.cameron_erdos :
    ∀ᶠ n : ℕ in atTop, (2 : ℝ) ^ ((n : ℝ) / 4) < fm n := by
…
```
Mentions: Erdos877.fm
theorem Erdos877.erdos_877.variants.balogh_liu_sharifzadeh_treglown Balogh, Liu, Sharifzadeh, and Treglown [BLST15] proved that $f_m(n)=2^{(\frac{1}{4}+o(1))n}$ sorry line 88 source
Source documentation
Balogh, Liu, Sharifzadeh, and Treglown [BLST15] proved that $f_m(n)=2^{(\frac{1}{4}+o(1))n}$.

depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Real.logb, OfNat.ofNat, Erdos877.fm, 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 erdos_877.variants.balogh_liu_sharifzadeh_treglown :
    Tendsto (fun n : ℕ ↦ logb 2 (fm n) / n) atTop (nhds (1 / 4)) := by
…
```
Mentions: Erdos877.fm