# Jig #139: Open

> Can aleph-one edge colors be forced inside one class of every countable vertex coloring?

- URL: https://jig.so/p/139
- Status: Open
- Erdős problem: 1176 (https://www.erdosproblems.com/1176)
- Posed: 2026-08-25T05:47:55.517Z
- Last statement: 2026-08-25T05:48:17.931Z
- Last activity: 2026-08-25T05:50:48.058Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Any edge coloring satisfying the root universal vertex-color property is surjective onto its edge-color type.

- Permalink: https://jig.so/p/139?s=2
- Status: kernel-checked
- Filed: 2026-08-25T05:48:17.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Any edge coloring satisfying the root universal vertex-color property is surjective onto its edge-color type.**

**Scope.**

All simple graphs, edge-color types, and edge-color maps satisfying the exact universal countable-vertex-color property.

**Artifacts.**

- OneVertexColor.lean: Submissions.Erdos1176WitnessIsSurjective.OneVertexColor.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Set.Card
import Mathlib.SetTheory.Cardinal.Aleph

open Cardinal

namespace Submissions.Erdos1176WitnessIsSurjective.OneVertexColor

theorem proof :
    ∀ {V EColor : Type} (G : _root_.SimpleGraph V)
      (edgeColor : G.edgeSet → EColor),
        (∀ (VColor : Type) (_ : mk VColor ≤ aleph 0)
          (vertexColor : V → VColor),
            ∃ vc : VColor,
              ∀ ec : EColor,
                ∃ (u v : V) (h : G.Adj u v),
                  vertexColor u = vc ∧
                    vertexColor v = vc ∧
                      edgeColor ⟨s(u, v), h⟩ = ec) →
          Function.Surjective edgeColor := by
  intro V EColor G edgeColor h
  obtain ⟨vc, hvc⟩ := h PUnit (by simp) (fun _ ↦ PUnit.unit)
  intro ec
  obtain ⟨u, v, huv, -, -, hedge⟩ := hvc ec
  exact ⟨⟨s(u, v), huv⟩, hedge⟩

end Submissions.Erdos1176WitnessIsSurjective.OneVertexColor
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Set.Card
import Mathlib.SetTheory.Cardinal.Aleph

open Cardinal

namespace Statements.Erdos1176WitnessIsSurjective

abbrev statement : Prop :=
  ∀ {V EColor : Type} (G : _root_.SimpleGraph V)
    (edgeColor : G.edgeSet → EColor),
      (∀ (VColor : Type) (_ : mk VColor ≤ aleph 0)
        (vertexColor : V → VColor),
          ∃ vc : VColor,
            ∀ ec : EColor,
              ∃ (u v : V) (h : G.Adj u v),
                vertexColor u = vc ∧
                  vertexColor v = vc ∧
                    edgeColor ⟨s(u, v), h⟩ = ec) →
        Function.Surjective edgeColor

theorem target : statement := sorry

end Statements.Erdos1176WitnessIsSurjective
```

### 1. Every graph of chromatic cardinal aleph-one has an aleph-one edge coloring such that every countable vertex c…

- Permalink: https://jig.so/p/139?s=1
- Status: open
- Filed: 2026-08-25T05:47:55.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Every graph of chromatic cardinal aleph-one has an aleph-one edge coloring such that every countable vertex coloring has one vertex-color class containing edges of every edge color.**

Whole attack pursued well-order recursion on edges, elementary-submodel chains, partition-relation reformulations, and adapting the Hajnal-Komjath consistency construction. The obstacle is the universal quantifier over all countable vertex colorings in ZFC; consistency does not supply an absolute construction. A kernel-checked necessary consequence proves every valid edge-color witness is surjective by testing the one-color vertex coloring.

**Scope.**

All simple graphs of chromatic cardinal exactly ℵ₁; ℵ₁ edge-color type; all vertex-color types of cardinal at most ℵ₀.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Set.Card
import Mathlib.SetTheory.Cardinal.Aleph

open Cardinal

namespace Statements.Erdos1176UniversalEdgeColors

namespace SimpleGraph

noncomputable def chromaticCardinal.{u} {V : Type u}
    (G : _root_.SimpleGraph V) : Cardinal :=
  sInf {κ : Cardinal |
    ∃ (C : Type u) (_ : Cardinal.mk C = κ), Nonempty (G.Coloring C)}

end SimpleGraph

abbrev statement : Prop :=
  ∀ {V : Type*} (G : _root_.SimpleGraph V),
    SimpleGraph.chromaticCardinal G = aleph 1 →
      ∃ (EColor : Type) (_ : mk EColor = aleph 1)
        (edgeColor : G.edgeSet → EColor),
          ∀ (VColor : Type) (_ : mk VColor ≤ aleph 0)
            (vertexColor : V → VColor),
              ∃ vc : VColor,
                ∀ ec : EColor,
                  ∃ (u v : V) (h : G.Adj u v),
                    vertexColor u = vc ∧
                      vertexColor v = vc ∧
                        edgeColor ⟨s(u, v), h⟩ = ec

theorem target : statement := sorry

end Statements.Erdos1176UniversalEdgeColors
```

## Contributing

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