FormalConjectures.ErdosProblems.«163»

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module

public import FormalConjecturesUtil

Erdős Problem 163

References:

erdosproblems.com/163

[BuEr75] Burr, S. A. and Erdős, P., On the Ramsey number of graphs with small degree-ratio. Colloq. Math. Soc. János Bolyai (1975).

[Le17] Lee, C., Ramsey numbers of degenerate graphs. Ann. of Math. (2) 185 (2017), 791-829.

@[expose] public sectionnamespace Erdos163

The Burr-Erdős conjecture: For any $d\geq 1$ if $H$ is a graph such that every subgraph contains a vertex of degree at most $d$ then $$R(H)\ll_d n.$$

Solved by Lee [Le17], who proved that $R(H) \leq 2^{2^{O(d)}}n$.

This problem is #9 in Ramsey Theory in the graphs problem collection.

@[category research solved, AMS 5]
theorem erdos_163 : answer(True) ↔
    ∀ (d : ℕ), 1 ≤ d →
      ∃ C > (0 : ℝ), ∀ (V : Type) [Fintype V] (H : SimpleGraph V),
        H.IsDegenerate d →
        (SimpleGraph.diagonalGraphRamsey H : ℝ) ≤ C * Fintype.card V := by⊢ True ↔
  ∀ (d : ℕ),
    1 ≤ d →
      ∃ C > 0,
        ∀ (V : Type) [inst : Fintype V] (H : SimpleGraph V),
          H.IsDegenerate d → ↑H.diagonalGraphRamsey ≤ C * ↑(Fintype.card V)
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos163