FormalConjectures.ErdosProblems.«547»

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module

public import FormalConjecturesUtil

Erdős Problem 547

References:

erdosproblems.com/547

[Bu74] Burr, S. A., Generalized Ramsey theory for graphs—a survey. Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973) (1974), 52-75.

[Zh11] Zhao, Y., Proof of the $(n/2-n/2-n/2)$ conjecture for large $n$. Electron. J. Combin. (2011), Paper 27, 61.

@[expose] public sectionopen Filternamespace Erdos547

If $T$ is a tree on $n$ vertices then $$R(T) \leq 2n-2.$$

This problem is #14 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_547 :
    ∀ (n : ℕ) (hn : 2 ≤ n) (T : SimpleGraph (Fin n)),
      T.IsTree → SimpleGraph.diagonalGraphRamsey T ≤ 2 * n - 2 := by⊢ ∀ (n : ℕ), 2 ≤ n → ∀ (T : SimpleGraph (Fin n)), T.IsTree → T.diagonalGraphRamsey ≤ 2 * n - 2
  sorryAll goals completed! 🐙

For sufficiently large $n$, every tree $T$ on $n$ vertices satisfies $R(T)\leq 2n-2$. Proved by Zhao [Zh11].

@[category research solved, AMS 5]
theorem erdos_547.variants.sufficiently_large :
    ∀ᶠ n : ℕ in atTop, ∀ T : SimpleGraph (Fin n),
      T.IsTree → SimpleGraph.diagonalGraphRamsey T ≤ 2 * n - 2 := by⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (T : SimpleGraph (Fin n)), T.IsTree → T.diagonalGraphRamsey ≤ 2 * n - 2
  sorryAll goals completed! 🐙

end Erdos547