FormalConjectures.ErdosProblems.«166»

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module

public import FormalConjecturesUtil

Erdős Problem 166

References:

erdosproblems.com/166

[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.

[AKS80] Ajtai, M., Komlós, J. and Szemerédi, E., A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.

[MaVe23] Mattheus, S. and Verstraëte, J., The asymptotics of $r(4,t)$. Ann. of Math. (2024), 941-965.

@[expose] public sectionopen Filternamespace Erdos166

Prove that $$R(4,k) \gg \frac{k^3}{(\log k)^{O(1)}}.$$

This is true, and was proved by Mattheus and Verstraëte [MaVe23], who showed that $R(4,k) \gg \frac{k^3}{(\log k)^4}$. This problem is #5 in Ramsey Theory in the graphs problem collection.

@[category research solved, AMS 5]
theorem erdos_166 : answer(True) ↔
    ∃ (c C : ℝ), 0 < c ∧ 0 < C ∧
      ∀ᶠ (k : ℕ) in atTop,
        (SimpleGraph.classicalRamsey 4 k : ℝ) ≥
          C * (k : ℝ) ^ 3 / (Real.log k) ^ c := by⊢ True ↔ ∃ c C, 0 < c ∧ 0 < C ∧ ∀ᶠ (k : ℕ) in atTop, ↑(SimpleGraph.classicalRamsey 4 k) ≥ C * ↑k ^ 3 / Real.log ↑k ^ c
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos166