FormalConjectures.ErdosProblems.«569»

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module

public import FormalConjecturesUtil

Erdős Problem 569

References:

erdosproblems.com/569

[EFRS93] Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.

@[expose] public sectionnamespace Erdos569

Let $k\geq 1$. What is the best possible $c_k$ such that $$R(C_{2k+1},H)\leq c_k m$$ for any graph $H$ on $m$ edges without isolated vertices?

This problem is #34 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_569 :
    let c : ℕ → ℝ := answer(sorry)
    ∀ (k : ℕ) (hk : 1 ≤ k),
      sInf {C : ℝ | 0 < C ∧
        ∀ (m : ℕ) (W : Type) [Fintype W] (H : SimpleGraph W) [DecidableRel H.Adj],
          (∀ v, 0 < H.degree v) →
          H.edgeSet.ncard = m →
          (SimpleGraph.graphRamsey (SimpleGraph.cycleGraph (2 * k + 1)) H : ℝ) ≤ C * m} =
            c k := by⊢ let c := sorry;
∀ (k : ℕ),
  1 ≤ k →
    sInf
        {C |
          0 < C ∧
            ∀ (m : ℕ) (W : Type) [inst : Fintype W] (H : SimpleGraph W) [inst_1 : DecidableRel H.Adj],
              (∀ (v : W), 0 < H.degree v) →
                H.edgeSet.ncard = m → ↑((SimpleGraph.cycleGraph (2 * k + 1)).graphRamsey H) ≤ C * ↑m} =
      c k
  sorryAll goals completed! 🐙

end Erdos569