82.lean`

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

4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 82

Reference: erdosproblems.com/82

Imports (1)

FormalConjecturesUtil

def Erdos82.IsRegularInduced A predicate that holds if $S$ is a regular induced subgraph of $G$ line 37 source
Source documentation
A predicate that holds if $S$ is a regular induced subgraph of $G$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, SimpleGraph.Subgraph
Proof uses: And, SimpleGraph.Subgraph.IsInduced, Exists, Nat, SimpleGraph.IsRegularOfDegree, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, Classical.propDecidable, SimpleGraph.Subgraph.Adj, Membership.mem, Set, SimpleGraph.neighborSet, SimpleGraph.Adj
7 instances, coercions and auxiliary constants left out.
```
def IsRegularInduced {G : SimpleGraph V} (S : Subgraph G) : Prop :=
…
```
def Erdos82.F $F(n)$ is the maximal integer such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices line 45 source
Source documentation
$F(n)$ is the maximal integer such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, SimpleGraph, Fin, Exists, SimpleGraph.Subgraph, And, Erdos82.IsRegularInduced, LE.le, Set.ncard, SimpleGraph.Subgraph.verts
3 instances, coercions and auxiliary constants left out.
```
noncomputable def F (n : ℕ) : ℕ :=
…
```
theorem Erdos82.erdos_82 $F(n) / \log n \to \infty as n \to \infty$ sorry line 53 source
Source documentation
$F(n) / \log n \to \infty as n \to \infty$

depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos82.F, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_82 : Tendsto (fun n => F n / Real.log n) atTop atTop := by
…
```
Mentions: Erdos82.F
theorem Erdos82.erdos_82.variants.F_upper_bound $F(n) \le O(n^{1/2} \ln ^ {3/4} n)$ Theorem 1.4 from [AKS07] [AKS07] Alon, N sorry line 64 source
Source documentation
$F(n) \le O(n^{1/2} \ln ^ {3/4} n)$
Theorem 1.4 from [AKS07]
[AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, Erdos82.F, HMul.hMul, Real.sqrt, HPow.hPow, Real.log, HDiv.hDiv, OfNat.ofNat
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_82.variants.F_upper_bound :
    (fun n => (F n : ℝ)) =O[atTop] (fun n => Real.sqrt n * (Real.log n) ^ (3 / 4 : ℝ)) := by
…
```
Mentions: Erdos82.F