600.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 600

Reference:

erdosproblems.com/600
erdosproblems.com/80
[Er87] Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.
[RuSz78] Ruzsa, I. Z. and Szemerédi, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II (1978), 939-945.

Imports (1)

FormalConjecturesUtil

def Erdos600.Erdos600Prop Documented declaration: Erdos600Prop line 42 source
Source documentation
Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SimpleGraph, Fin, GE.ge, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, Membership.mem, Finset, Finset.Nonempty, SimpleGraph.trianglesContaining, Exists, And, LE.le
6 instances, coercions and auxiliary constants left out.
```
def Erdos600Prop (n : ℕ) (e : ℕ) (r : ℕ) : Prop :=
…
```
def Erdos600.eFunction line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Erdos600.Erdos600Prop
1 instances, coercions and auxiliary constants left out.
theorem Erdos600.erdos_600.parts.i Let $r \geq 2$ sorry line 53 source
Source documentation
Let $r \geq 2$. Is it true that $e(n,r+1) - e(n,r) \to \infty$ as $n \to \infty$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Filter.Tendsto, Real, HSub.hSub, Erdos600.eFunction, HAdd.hAdd, Filter.atTop
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_600.parts.i :
    answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →
      Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) - (eFunction n r : ℝ)) atTop atTop := by
…
```
Mentions: Erdos600.eFunction
theorem Erdos600.erdos_600.parts.ii Let $r \geq 2$ sorry line 62 source
Source documentation
Let $r \geq 2$. Is it true that $\frac{e(n,r+1)}{e(n,r)} \to 1$ as $n \to \infty$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Filter.Tendsto, Real, HDiv.hDiv, Erdos600.eFunction, HAdd.hAdd, 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_600.parts.ii :
    answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →
      Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) / (eFunction n r : ℝ)) atTop (𝓝 1) := by
…
```
Mentions: Erdos600.eFunction
theorem Erdos600.erdos_600.variants.ruzsa_szemeredi_upper_bound Ruzsa and Szemerédi [RuSz78] proved that $e(n,r)=o(n^2)$ for any fixed $r$ sorry line 71 source
Source documentation
Ruzsa and Szemerédi [RuSz78] proved that $e(n,r)=o(n^2)$ for any fixed $r$.

depends on sorry what this means
Statement uses: Nat, Asymptotics.IsLittleO, Real, Filter.atTop, Erdos600.eFunction, HPow.hPow, OfNat.ofNat
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_600.variants.ruzsa_szemeredi_upper_bound :
    ∀ r : ℕ, (fun (n : ℕ) ↦ (eFunction n r : ℝ)) =o[atTop] (fun (n : ℕ) ↦ (n : ℝ)^2) := by
…
```
Mentions: Erdos600.eFunction