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import FormalConjecturesUtilErdős Problem 1104
namespace Erdos1104
open Filter SimpleGraphopen scoped Real
Maximum chromatic number of a triangle-free graph on n vertices.
noncomputable def triangleFreeMaxChromatic (n : ℕ) : ℕ :=
sSup {χ | ∃ G : SimpleGraph (Fin n), G.CliqueFree 3 ∧ G.chromaticNumber = χ}
Lower bound (Hefty–Horn–King–Pfender 2025).
There exists a constant $c_1 \in (0,1]$ such that, for sufficiently large $n$,
$$
c_1 \sqrt{\frac{n}{\log n}} \le f(n),
$$
where $f(n)$ denotes the maximum chromatic number of a triangle-free graph on
$n$ vertices, formalized as triangleFreeMaxChromatic n.
@[category research solved, AMS 5]
theorem erdos_1104.variants.lower :
∃ c₁ : ℝ, 0 < c₁ ∧ c₁ ≤ 1 ∧
(∀ᶠ n : ℕ in atTop,
c₁ * Real.sqrt (n : ℝ) / Real.sqrt (Real.log (n : ℝ))
≤ (triangleFreeMaxChromatic n : ℝ)) := ⊢ ∃ c₁, 0 < c₁ ∧ c₁ ≤ 1 ∧ ∀ᶠ (n : ℕ) in atTop, c₁ * √↑n / √(Real.log ↑n) ≤ ↑(triangleFreeMaxChromatic n)
All goals completed! 🐙
Upper bound (Davies–Illingworth 2022).
There exists a constant $c_2 \ge 2$ such that, for sufficiently large $n$,
$$
f(n) \le c_2 \sqrt{\frac{n}{\log n}},
$$
where $f(n)$ denotes the maximum chromatic number of a triangle-free graph on
$n$ vertices, formalized as triangleFreeMaxChromatic n.
@[category research solved, AMS 5]
theorem erdos_1104.variants.upper :
∃ c₂ : ℝ, 2 ≤ c₂ ∧
(∀ᶠ n : ℕ in atTop,
(triangleFreeMaxChromatic n : ℝ)
≤ c₂ * Real.sqrt (n : ℝ) / Real.sqrt (Real.log (n : ℝ))) := ⊢ ∃ c₂, 2 ≤ c₂ ∧ ∀ᶠ (n : ℕ) in atTop, ↑(triangleFreeMaxChromatic n) ≤ c₂ * √↑n / √(Real.log ↑n)
All goals completed! 🐙
end Erdos1104