FormalConjectures.ErdosProblems.«159»

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module

public import FormalConjecturesUtil

Erdős Problem 159

References:

erdosproblems.com/159

[Er78] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Proc. Ninth Southeastern Conf. Combinatorics, Graph Theory and Computing (1978), 29-40.

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

@[expose] public sectionnamespace Erdos159

There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$

The prize of $100 is offered in [Er78] for a proof or disproof.

This problem is #17 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_159 :
    ∃ (c : ℝ) (_ : 0 < c) (C : ℝ),
      ∀ (n : ℕ), 1 ≤ n →
        (SimpleGraph.graphRamsey (SimpleGraph.cycleGraph 4)
          (SimpleGraph.completeGraph (Fin n)) : ℝ) ≤ C * (n : ℝ) ^ (2 - c) := by⊢ ∃ c,
  ∃ (_ : 0 < c),
    ∃ C,
      ∀ (n : ℕ),
        1 ≤ n → ↑((SimpleGraph.cycleGraph 4).graphRamsey (SimpleGraph.completeGraph (Fin n))) ≤ C * ↑n ^ (2 - c)
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos159