FormalConjectures.ErdosProblems.«986»

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module

public import FormalConjecturesUtil

Erdős Problem 986

References:

erdosproblems.com/986

[ChGr98] Chung, F. and Graham, R., Erdős on Graphs: His Legacy of Unsolved Problems. A K Peters, Ltd. (1998).

[Br26] Bradač, D., Off-diagonal Ramsey numbers. arXiv:2605.28793 (2026).

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

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

@[expose] public sectionopen Filternamespace Erdos986

For any fixed $s\geq 3$, $$R(s,k) \gg \frac{k^{s-1}}{(\log k)^c}$$ for some constant $c=c(s)>0$.

According to Chung and Graham [ChGr98] this was first conjectured by Erdős in 1947.

Proved by Bradač [Br26], with $c=2s-4$.

@[category research solved, AMS 5]
theorem erdos_986 :
    ∀ (s : ℕ) (hs : 3 ≤ s),
      ∃ (c C : ℝ), 0 < c ∧ 0 < C ∧
        ∀ᶠ (k : ℕ) in atTop,
          (SimpleGraph.classicalRamsey s k : ℝ) ≥
            C * (k : ℝ) ^ (s - 1) / (Real.log k) ^ c := by⊢ ∀ (s : ℕ),
  3 ≤ s →
    ∃ c C, 0 < c ∧ 0 < C ∧ ∀ᶠ (k : ℕ) in atTop, ↑(SimpleGraph.classicalRamsey s k) ≥ C * ↑k ^ (s - 1) / Real.log ↑k ^ c
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos986