FormalConjectures.ErdosProblems.«546»

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module

public import FormalConjecturesUtil

Erdős Problem 546

References:

erdosproblems.com/546

[Su11] Sudakov, B., A conjecture of Erdős on graph Ramsey numbers. Adv. Math. (2011), 3148-3155.

[AKS03] Alon, N., Krivelevich, M. and Sudakov, B., Turán numbers of bipartite graphs and Ramsey graphs of bounded degree. Combin. Probab. Comput. (2003), 477-483.

@[expose] public sectionnamespace Erdos546

Let $G$ be a graph with no isolated vertices and $m$ edges. Is it true that $$R(G) \leq 2^{O(m^{1/2})}?$$

This is true, and was proved by Sudakov [Su11].

This problem is #11 in Ramsey Theory in the graphs problem collection.

@[category research solved, AMS 5]
theorem erdos_546 : answer(True) ↔
    ∃ C > (0 : ℝ), ∀ (m : ℕ) (V : Type) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],
      (∀ v, 0 < G.degree v) →
      G.edgeSet.ncard = m →
      (SimpleGraph.diagonalGraphRamsey G : ℝ) ≤ 2 ^ (C * Real.sqrt m) := by⊢ True ↔
  ∃ C > 0,
    ∀ (m : ℕ) (V : Type) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],
      (∀ (v : V), 0 < G.degree v) → G.edgeSet.ncard = m → ↑G.diagonalGraphRamsey ≤ 2 ^ (C * √↑m)
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos546