FormalConjectures.ErdosProblems.«550»

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module

public import FormalConjecturesUtil

Erdős Problem 550

References:

erdosproblems.com/550

[Ch77] Chvátal, V., Tree-complete graph Ramsey numbers. J. Graph Theory (1977), 93.

[Li26] Li, E., A Resolution of Erdős Problem 550 on Tree versus Complete Multipartite Ramsey Numbers, arXiv:2606.23659 (2026).

@[expose] public sectionopen Filternamespace Erdos550

Let $m_1\leq\cdots\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\ldots,m_k$ then prove that $$R(T,G)\leq (\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.$$

This problem is #16 in Ramsey Theory in the graphs problem collection.

Li [Li26] proved this, combining an off-Turán tree-embedding theorem with a compactness theorem for bounded-rank hypergraph obstructions. The linked formal proof states the bound for trees on an arbitrary finite vertex type and with K_{m_1,m_2} written as a complete multipartite graph with two parts; the statement below is the special case V = Fin n.

@[category research solved, AMS 5, formal_proof using lean4 at
  "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos550/Main.lean#L80"]
theorem erdos_550 :
    ∀ (k : ℕ) (hk : 2 ≤ k) (m : Fin k → ℕ) (hm : Monotone m)
      (hm_pos : ∀ i, 0 < m i),
      ∀ᶠ n : ℕ in atTop,
        ∀ (T : SimpleGraph (Fin n)), T.IsTree →
          SimpleGraph.graphRamsey T
            (SimpleGraph.completeMultipartiteGraph (fun i ↦ Fin (m i))) ≤
            (k - 1) * (SimpleGraph.graphRamsey T
              (completeBipartiteGraph (Fin (m ⟨0, byk:ℕhk:2 ≤ km:Fin k → ℕhm:Monotone mhm_pos:∀ (i : Fin k), 0 < m in:ℕT:SimpleGraph (Fin n)⊢ 0 < k omegaAll goals completed! 🐙⟩)) (Fin (m ⟨1, byk:ℕhk:2 ≤ km:Fin k → ℕhm:Monotone mhm_pos:∀ (i : Fin k), 0 < m in:ℕT:SimpleGraph (Fin n)⊢ 1 < k omegaAll goals completed! 🐙⟩))) - 1) +
                m ⟨0, byk:ℕhk:2 ≤ km:Fin k → ℕhm:Monotone mhm_pos:∀ (i : Fin k), 0 < m in:ℕT:SimpleGraph (Fin n)⊢ 0 < k omegaAll goals completed! 🐙⟩ := by⊢ ∀ (k : ℕ) (hk : 2 ≤ k) (m : Fin k → ℕ),
  Monotone m →
    (∀ (i : Fin k), 0 < m i) →
      ∀ᶠ (n : ℕ) in atTop,
        ∀ (T : SimpleGraph (Fin n)),
          T.IsTree →
            T.graphRamsey (SimpleGraph.completeMultipartiteGraph fun i ↦ Fin (m i)) ≤
              (k - 1) * (T.graphRamsey (completeBipartiteGraph (Fin (m ⟨0, ⋯⟩)) (Fin (m ⟨1, ⋯⟩))) - 1) + m ⟨0, ⋯⟩
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos550