FormalConjectures.ErdosProblems.«1176»

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import FormalConjecturesUtil

Erdős Problem 1176

Reference: erdosproblems.com/1176


open Cardinal


namespace Erdos1176

Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?

A problem of Erdős, Galvin, and Hajnal. The consistency of this was proved by Hajnal and Komjáth.

@[category research open, AMS 3 5]
theorem declaration uses 'sorry'erdos_1176 :
    answer(sorry) ↔ ∀ {V : Type*} (G : SimpleGraph V), G.chromaticCardinal = aleph 1 →
      ∃ (EColor : Type) (_ : mk EColor = aleph 1) (c_edge : G.edgeSet → EColor),
        ∀ (VColor : Type) (_ : mk VColor ≤ aleph 0) (c_vert : V → VColor),
          ∃ (vc : VColor),
            ∀ (ec : EColor), ∃ (u v : V) (h : G.Adj u v),
              c_vert u = vc ∧ c_vert v = vc ∧ c_edge ⟨s(u, v), h⟩ = ec := by⊢ True ↔
  ∀ {V : Type u_1} (G : SimpleGraph V),
    G.chromaticCardinal = ℵ_ 1 →
      ∃ EColor,
        ∃ (_ : #EColor = ℵ_ 1),
          ∃ c_edge,
            ∀ (VColor : Type),
              #VColor ≤ ℵ_ 0 →
                ∀ (c_vert : V → VColor),
                  ∃ vc,
                    ∀ (ec : EColor), ∃ u v, ∃ (h : G.Adj u v), c_vert u = vc ∧ c_vert v = vc ∧ c_edge ⟨s(u, v), h⟩ = ec
  sorryAll goals completed! 🐙


end Erdos1176