FormalConjectures.ErdosProblems.«1030»

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module

public import FormalConjecturesUtil

Erdős Problem 1030

References:

erdosproblems.com/1030

[BEFS89] Burr, S. and Erdős, P. and Faudree, R. J. and Schelp, R. H., On the difference between consecutive Ramsey numbers. Utilitas Math. (1989), 115-118.

@[expose] public sectionopen Filternamespace Erdos1030

Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$.

Prove the existence of some $c>0$ such that $$\lim_{k\to \infty}\frac{R(k+1,k)}{R(k,k)}> 1+c.$$

A problem of Erdős and Sós.

@[category research open, AMS 5]
theorem erdos_1030 :
    ∃ c > (0 : ℝ), ∃ L : ℝ,
      Tendsto (fun k : ℕ ↦
        (SimpleGraph.classicalRamsey (k + 1) k : ℝ) /
          (SimpleGraph.classicalRamsey k k : ℝ)) atTop (nhds L) ∧
      L > 1 + c := by⊢ ∃ c > 0,
  ∃ L,
    Tendsto (fun k ↦ ↑(SimpleGraph.classicalRamsey (k + 1) k) / ↑(SimpleGraph.classicalRamsey k k)) atTop (nhds L) ∧
      L > 1 + c
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos1030