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import FormalConjecturesUtilErdős Problem 920
[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.
[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).
open Real Filter
namespace Erdos920
$f_k(n)$ is the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.
noncomputable def f (k n : ℕ) : ℕ :=
sSup {(G.chromaticNumber) | (G : SimpleGraph (Fin n)) (_ : G.CliqueFree k)}
Is it true that, for $k\geq 4$, $f_k(n) \gg \frac{n^{1-\frac{1}{k-1}}}{(\log n)^{c_k}}$ for some constant $c_k>0$?
@[category research open, AMS 5]
theorem erdos_920 :
answer(sorry) ↔ ∀ k : ℕ, k ≥ 4 → ∃ c > 0,
(fun n ↦ f k n) ≫ (fun n ↦ (n : ℝ) ^ (1 - 1 / ((k : ℝ) - 1)) / (log n) ^ c) := ⊢ True ↔ ∀ k ≥ 4, ∃ c > 0, (fun n => ↑n ^ (1 - 1 / (↑k - 1)) / log ↑n ^ c) =O[atTop] fun n => ↑(f k n)
All goals completed! 🐙
Graver and Yackel [GrYa68] proved that $f_k(n) \ll \left(n\frac{\log\log n}{\log n}\right)^{1-\frac{1}{k-1}}.$
@[category research solved, AMS 5]
theorem erdos_920.variants.upper_bound (k : ℕ) (hk : k ≥ 3) :
(fun n ↦ f k n) ≪ (fun n ↦ ((n : ℝ) * log (log n) / log n) ^ (1 - 1 / ((k : ℝ) - 1))) := k:ℕhk:k ≥ 3⊢ (fun n => ↑(f k n)) =O[atTop] fun n => (↑n * log (log ↑n) / log ↑n) ^ (1 - 1 / (↑k - 1))
All goals completed! 🐙
It is known that $f_3(n)\asymp (n/\log n)^{1/2}$ (see [erdosproblems.com/1104]).
@[category research solved, AMS 5]
theorem erdos_920.variants.k_eq_3 :
(fun n ↦ (f 3 n : ℝ)) =Θ[atTop] (fun n ↦ ((n : ℝ) / log n) ^ (1 / 2 : ℝ)) := ⊢ (fun n => ↑(f 3 n)) =Θ[atTop] fun n => (↑n / log ↑n) ^ (1 / 2)
All goals completed! 🐙
The lower bound $R(4,m) \gg m^3/(\log m)^4$ of Mattheus and Verstraete [MaVe23] (see [erdosproblems.com/166]) implies $f_4(n) \gg \frac{n^{2/3}}{(\log n)^{4/3}}$.
@[category research solved, AMS 5]
theorem erdos_920.variants.lower_bound_f4 :
(fun n ↦ f 4 n) ≫ (fun n ↦ (n : ℝ) ^ (2 / 3 : ℝ) / (log n) ^ (4 / 3 : ℝ)) := ⊢ (fun n => ↑n ^ (2 / 3) / log ↑n ^ (4 / 3)) =O[atTop] fun n => ↑(f 4 n)
All goals completed! 🐙
A positive answer to this question would follow from [erdosproblems.com/986]. The known bounds for that problem imply $f_k(n) \gg \frac{n^{1-\frac{2}{k+1}}}{(\log n)^{c_k}}.$
@[category research solved, AMS 5]
theorem erdos_920.variants.lower_bound (k : ℕ) (hk : k ≥ 3) :
∃ c > 0, (fun n ↦ f k n) ≫ (fun (n : ℕ) ↦
(n : ℝ) ^ (1 - 2 / ((k : ℝ) + 1)) / (log n) ^ c) := k:ℕhk:k ≥ 3⊢ ∃ c > 0, (fun n => ↑n ^ (1 - 2 / (↑k + 1)) / log ↑n ^ c) =O[atTop] fun n => ↑(f k n)
All goals completed! 🐙
end Erdos920