FormalConjectures.ErdosProblems.«544»

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module

public import FormalConjecturesUtil

Erdős Problem 544

References:

erdosproblems.com/544

@[expose] public sectionopen Filternamespace Erdos544

Show that $$R(3,k+1)-R(3,k)\to\infty$$ as $k\to \infty$.

A problem of Erdős and Sós. This problem is #8 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_544.parts.i :
    Tendsto (fun k : ℕ ↦ (SimpleGraph.classicalRamsey 3 (k + 1) : ℝ) -
      (SimpleGraph.classicalRamsey 3 k : ℝ)) atTop atTop := by⊢ Tendsto (fun k ↦ ↑(SimpleGraph.classicalRamsey 3 (k + 1)) - ↑(SimpleGraph.classicalRamsey 3 k)) atTop atTop
  sorryAll goals completed! 🐙

Similarly, prove or disprove that $$R(3,k+1)-R(3,k)=o(k).$$

@[category research open, AMS 5]
theorem erdos_544.parts.ii : answer(sorry) ↔
    (fun k : ℕ ↦ (SimpleGraph.classicalRamsey 3 (k + 1) : ℝ) -
      (SimpleGraph.classicalRamsey 3 k : ℝ)) =o[atTop] (fun k : ℕ ↦ (k : ℝ)) := by⊢ True ↔ (fun k ↦ ↑(SimpleGraph.classicalRamsey 3 (k + 1)) - ↑(SimpleGraph.classicalRamsey 3 k)) =o[atTop] fun k ↦ ↑k
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos544