FormalConjectures.ErdosProblems.«753»

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import FormalConjecturesUtil

Erdős Problem 753

References:

erdosproblems.com/753

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

namespace Erdos753

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.

def IsKChoosable {V : Type*} (G : SimpleGraph V) (k : ℕ) : Prop :=
  ∀ L : V → Finset ℕ, (∀ v, (L v).card = k) → ∃ C : G.Coloring ℕ, ∀ v, C v ∈ L v

The list chromatic number $\chi_L(G)$, defined to be the minimal $k$ such that $G$ is $k$-choosable.

noncomputable def listChromaticNumber {V : Type*} (G : SimpleGraph V) : ℕ :=
  sInf {k : ℕ | IsKChoosable G k}

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.

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos753.lean"]
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⊢ False ↔
  ∃ c,
    0 < c ∧
      ∀ (n : ℕ),
        0 < n → ∀ (G : SimpleGraph (Fin n)), ↑n ^ (1 / 2 + c) < ↑(listChromaticNumber G) + ↑(listChromaticNumber Gᶜ)
  sorryAll goals completed! 🐙

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.

@[category research solved, AMS 5]
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⊢ ∃ C,
  0 < C ∧
    ∀ (n : ℕ), 2 ≤ n → ∃ G, ↑(listChromaticNumber G) + ↑(listChromaticNumber Gᶜ) ≤ C * (↑n * Real.log ↑n) ^ (1 / 2)
  sorryAll goals completed! 🐙

end Erdos753