1176.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1176

Reference: erdosproblems.com/1176

Imports (1)

FormalConjecturesUtil

theorem Erdos1176.erdos_1176 Let $G$ be a graph with chromatic number $\aleph_1$ sorry line 40 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph, Eq, Cardinal, SimpleGraph.chromaticCardinal, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Exists, Cardinal.mk, Sym2, SimpleGraph.edgeSet, LE.le, SimpleGraph.Adj, And, Subtype.mk, Membership.mem, Set and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem 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
…
```