753.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 753

References:

erdosproblems.com/753
[Al92] Alon, Noga, Choice numbers of graphs: a probabilistic approach. Combin. Probab. Comput. (1992), 107-114.

Imports (1)

FormalConjecturesUtil

def Erdos753.IsKChoosable Documented declaration: IsKChoosable line 38 source
Source documentation
A graph $G$ is $k$-choosable if for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: Finset, Eq, Finset.card, Exists, SimpleGraph.Coloring, Membership.mem, SimpleGraph.Adj, SimpleGraph.completeGraph
4 instances, coercions and auxiliary constants left out.
```
def IsKChoosable {V : Type*} (G : SimpleGraph V) (k : ℕ) : Prop :=
…
```
def Erdos753.listChromaticNumber The list chromatic number $\chi_L(G)$, defined to be the minimal $k$ such that $G$ is $k$-choosable line 45 source
Source documentation
The list chromatic number $\chi_L(G)$, defined to be the minimal $k$ such that $G$ is $k$-choosable.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: InfSet.sInf, Set.ofPred, Erdos753.IsKChoosable
1 instances, coercions and auxiliary constants left out.
```
noncomputable def listChromaticNumber {V : Type*} (G : SimpleGraph V) : ℕ :=
…
```
theorem Erdos753.erdos_753 The list chromatic number sorry line 66 source
Source documentation
The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.
Does there exist some constant $c>0$ such that $$\chi_L(G)+\chi_L(G^c)> n^{1/2+c}$$ for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?
A problem of Erdős, Rubin, and Taylor.
The answer is no: Alon [Al92] proved that, for every $n$, there exists a graph $G$ on $n$ vertices such that $$\chi_L(G)+\chi_L(G^c)\ll (n\log n)^{1/2},$$ where the implied constant is absolute.

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, SimpleGraph, Fin, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Erdos753.listChromaticNumber, Compl.compl
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_753 : answer(False) ↔
    ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 0 < n → ∀ G : SimpleGraph (Fin n),
      (n : ℝ) ^ ((1 : ℝ) / 2 + c) <
        (listChromaticNumber G : ℝ) + (listChromaticNumber Gᶜ : ℝ) := by
…
```
Mentions: c, Erdos753.listChromaticNumber
theorem Erdos753.erdos_753.variants.alon Alon [Al92] proved that, for every $n$, there exists a graph $G$ on $n$ vertices such that $$\chi_L(G)+\chi_L(G^c)\ll (n\log n)^{1/2},$$ where the implied constant is absolute sorry line 78 source
Source documentation
Alon [Al92] proved that, for every $n$, there exists a graph $G$ on $n$ vertices such that $$\chi_L(G)+\chi_L(G^c)\ll (n\log n)^{1/2},$$ where the implied constant is absolute.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, SimpleGraph, Fin, HAdd.hAdd, Erdos753.listChromaticNumber, Compl.compl, HMul.hMul, HPow.hPow, Real.log, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_753.variants.alon :
    ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ, 2 ≤ n → ∃ G : SimpleGraph (Fin n),
      (listChromaticNumber G : ℝ) + (listChromaticNumber Gᶜ : ℝ) ≤
        C * ((n : ℝ) * Real.log (n : ℝ)) ^ ((1 : ℝ) / 2) := by
…
```
Mentions: Erdos753.listChromaticNumber