757.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 757

References:

erdosproblems.com/757
[GyLe95] Gyárfás, András and Lehel, Jenő, Linear sets with five distinct differences among any four elements. J. Combin. Theory Ser. B (1995), 108-118.

Imports (1)

FormalConjecturesUtil

def Erdos757.IsAdmissible Documented declaration: IsAdmissible line 40 source
Source documentation
We say that c is admissible if, for any finite set A such that every four-element subset B determines at least five positive differences, there exists a Sidon subset S of size at least c * A.ncard. The difference condition is equivalent to 11 ≤ (B - B).ncard.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Set, Set.Finite, LE.le, Eq, Nat, Set.ncard, OfNat.ofNat, HSub.hSub, Exists, And, IsSidon, HMul.hMul
12 instances, coercions and auxiliary constants left out.
```
def IsAdmissible (c : ℝ) : Prop := ∀ {A : Set ℝ}, A.Finite → (∀ B ⊆ A,
…
```
Mentions: c
theorem Erdos757.erdos_757 What is the supremum of the set of admissible numbers? sorry line 45 source
Source documentation
What is the supremum of the set of admissible numbers?

depends on sorry what this means
Statement uses: Set, Real, Eq, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat, SupSet.sSup, Set.ofPred, Erdos757.IsAdmissible
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_757 {A : Set ℝ} :
    answer(sorry) = sSup {c | IsAdmissible c} := by
…
```
Mentions: c, Erdos757.IsAdmissible
theorem Erdos757.erdos_757.variants.lowerBound The supremum is strictly larger than 1 / 2, which is proved in [GyLe95] sorry line 51 source
Source documentation
The supremum is strictly larger than 1 / 2, which is proved in [GyLe95].

depends on sorry what this means
Statement uses: Set, Real, LT.lt, HDiv.hDiv, OfNat.ofNat, Nat, SupSet.sSup, Set.ofPred, Erdos757.IsAdmissible
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_757.variants.lowerBound {A : Set ℝ} : 1 / (2 : ℝ) < sSup {c | IsAdmissible c} := by
…
```
Mentions: c, Erdos757.IsAdmissible
theorem Erdos757.erdos_757.variants.upperBound In [GyLe95], the authors also prove that the supremum is smaller than 3 / 5 sorry line 56 source
Source documentation
In [GyLe95], the authors also prove that the supremum is smaller than 3 / 5.

depends on sorry what this means
Statement uses: Set, Real, LT.lt, SupSet.sSup, Set.ofPred, Erdos757.IsAdmissible, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_757.variants.upperBound {A : Set ℝ} : sSup {c | IsAdmissible c} < 3 / (5 : ℝ) := by
…
```
Mentions: c, Erdos757.IsAdmissible