FormalConjectures.ErdosProblems.«800»

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module

public import FormalConjecturesUtil

Erdős Problem 800

References:

erdosproblems.com/800

[Al94] Alon, N., Subdivided graphs have linear Ramsey numbers. J. Graph Theory (1994), 343-347.

@[expose] public sectionnamespace Erdos800

If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $\geq 3$ then $$R(G)\ll n,$$ where the implied constant is absolute.

A problem of Burr and Erdős. Solved in the affirmative by Alon [Al94].

@[category research solved, AMS 5]
theorem erdos_800 : answer(True) ↔
    ∃ C > (0 : ℝ), ∀ (n : ℕ) (V : Type) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],
      Fintype.card V = n →
      (∀ u v, G.Adj u v → ¬(3 ≤ G.degree u ∧ 3 ≤ G.degree v)) →
      (SimpleGraph.diagonalGraphRamsey G : ℝ) ≤ C * n := by⊢ True ↔
  ∃ C > 0,
    ∀ (n : ℕ) (V : Type) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],
      Fintype.card V = n →
        (∀ (u v : V), G.Adj u v → ¬(3 ≤ G.degree u ∧ 3 ≤ G.degree v)) → ↑G.diagonalGraphRamsey ≤ C * ↑n
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos800