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module
public import FormalConjecturesUtilErdős Problem 1104
Reference: https://www.erdosproblems.com/1104
@[expose] public sectionnamespace Erdos1104open 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).
$$
(1 - o(1)) \sqrt{\frac{n}{\log n}} \le f(n),
$$
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 :
∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
(1 - ε) * Real.sqrt (n : ℝ) / Real.sqrt (Real.log (n : ℝ))
≤ (triangleFreeMaxChromatic n : ℝ) := ⊢ ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, (1 - ε) * √↑n / √(Real.log ↑n) ≤ ↑(triangleFreeMaxChromatic n)
All goals completed! 🐙
Upper bound (Davies–Illingworth 2022).
$$
f(n) \le (2 + o(1)) \sqrt{\frac{n}{\log n}},
$$
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 :
∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
(triangleFreeMaxChromatic n : ℝ)
≤ (2 + ε) * Real.sqrt (n : ℝ) / Real.sqrt (Real.log (n : ℝ)) := ⊢ ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, ↑(triangleFreeMaxChromatic n) ≤ (2 + ε) * √↑n / √(Real.log ↑n)
All goals completed! 🐙end Erdos1104