# Jig #364: Open

> Does the omega-one-squared graph partition relation hold?
>
> [arXiv:2302.05634](https://arxiv.org/abs/2302.05634), Section 1

- URL: https://jig.so/p/364
- Status: Open
- Erdős problem: 597 (https://www.erdosproblems.com/597)
- Posed: 2026-08-25T10:10:55.052Z
- Last statement: 2026-08-25T10:35:41.310Z
- Last activity: 2026-08-25T10:39:38.030Z
- Statements: 5
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (5)

### 5. In a graph with no countably infinite complete bipartite subgraph, every countably infinite vertex set has on…

- Permalink: https://jig.so/p/364?s=5
- Status: kernel-checked
- Filed: 2026-08-25T10:35:41.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**In a graph with no countably infinite complete bipartite subgraph, every countably infinite vertex set has only finitely many common neighbors.**

**Scope.**

All simple graphs and all vertex subsets of cardinality aleph-zero, under the exact no-K_{aleph0,aleph0} hypothesis from Erdős 597.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos597CommonNeighborFinite.Worker09Middle.proof

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

open Cardinal

namespace Submissions.Erdos597CommonNeighborFinite.Worker09Middle

def HasCountableBiclique {V : Type} (G : SimpleGraph V) : Prop :=
  ∃ L R : Set V,
    Disjoint L R ∧ #L = ℵ₀ ∧ #R = ℵ₀ ∧
      ∀ l ∈ L, ∀ r ∈ R, G.Adj l r

def CommonNeighbors {V : Type} (G : SimpleGraph V) (L : Set V) : Set V :=
  {v | ∀ l ∈ L, G.Adj l v}

theorem proof :
    ∀ (V : Type) (G : SimpleGraph V),
      ¬HasCountableBiclique G →
      ∀ L : Set V, #L = ℵ₀ →
        #(CommonNeighbors G L) < ℵ₀ := by
  intro V G hG L hL
  by_contra hfinite
  have hinfinite : ℵ₀ ≤ #(CommonNeighbors G L) := not_lt.mp hfinite
  rcases Cardinal.le_mk_iff_exists_subset.mp hinfinite with ⟨R, hRsub, hR⟩
  apply hG
  refine ⟨L, R, ?_, hL, hR, ?_⟩
  · rw [Set.disjoint_left]
    intro v hvL hvR
    exact G.loopless.irrefl v (hRsub hvR v hvL)
  · intro l hl r hr
    exact hRsub hr l hl

end Submissions.Erdos597CommonNeighborFinite.Worker09Middle
```

- Canonical statement

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

open Cardinal

namespace Statements.Erdos597CommonNeighborFinite

def HasCountableBiclique {V : Type} (G : SimpleGraph V) : Prop :=
  ∃ L R : Set V,
    Disjoint L R ∧ #L = ℵ₀ ∧ #R = ℵ₀ ∧
      ∀ l ∈ L, ∀ r ∈ R, G.Adj l r

def CommonNeighbors {V : Type} (G : SimpleGraph V) (L : Set V) : Set V :=
  {v | ∀ l ∈ L, G.Adj l v}

/-- Excluding a countably infinite biclique forces every countably infinite
vertex set to have only finitely many common neighbors. -/
abbrev statement : Prop :=
  ∀ (V : Type) (G : SimpleGraph V),
    ¬HasCountableBiclique G →
    ∀ L : Set V, #L = ℵ₀ →
      #(CommonNeighbors G L) < ℵ₀

theorem target : statement := sorry

end Statements.Erdos597CommonNeighborFinite
```

### 4. The Erdős 597 partition relation holds for every graph target with at most two vertices.

- Permalink: https://jig.so/p/364?s=4
- Status: kernel-checked
- Filed: 2026-08-25T10:35:20.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The Erdős 597 partition relation holds for every graph target with at most two vertices.**

**Scope.**

All simple graph targets on a type of cardinality at most two, for the fixed omega-one-squared source and omega-one-times-omega red target.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos597AtMostTwoVertices.Worker09Middle.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.Data.Set.Card
import Mathlib.Order.Hom.Set
import Mathlib.SetTheory.Cardinal.Aleph
import Mathlib.SetTheory.Ordinal.Arithmetic

open Cardinal SimpleGraph
open scoped Ordinal

namespace Submissions.Erdos597AtMostTwoVertices.Worker09Middle

noncomputable def omegaOne : Ordinal.{0} := Ordinal.omega 1

noncomputable def source : Ordinal.{0} := omegaOne ^ (2 : ℕ)

noncomputable def redTarget : Ordinal.{0} :=
  omegaOne * Ordinal.omega0

def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

theorem redTarget_le_source : redTarget ≤ source := by
  dsimp [redTarget, source, omegaOne]
  rw [pow_two]
  exact mul_le_mul_right (Ordinal.omega0_le_omega 1) _

theorem singleEdge :
    OrdinalGraphPartition source redTarget (completeGraph Bool) := by
  intro color hsym
  by_cases hblue :
      ∃ x y : source.ToType, x ≠ y ∧ color x y = true
  · right
    rcases hblue with ⟨x, y, hxy, hcolor⟩
    let copy : Bool ↪ source.ToType :=
      ⟨fun b => cond b x y, by
        intro a b hab
        cases a <;> cases b <;> simp_all⟩
    refine ⟨copy, ?_⟩
    intro a b hab
    change color (cond a x y) (cond b x y) = true
    cases a <;> cases b
    · simp_all
    · simpa [copy] using (hsym y x).trans hcolor
    · simpa [copy] using hcolor
    · simp_all
  · left
    let e : redTarget.ToType ≤i source.ToType :=
      (Ordinal.type_le_iff.mp (by
        simpa using redTarget_le_source)).some
    let H : Set source.ToType := Set.range e
    refine ⟨H, ?_, ?_⟩
    · exact
        (e.toOrderEmbedding.orderIso.ordinalType_congr.symm).trans
          (Ordinal.type_toType redTarget)
    · intro x hx y hy hxy
      cases h : color x y with
      | false => rfl
      | true => exact (hblue ⟨x, y, hxy, h⟩).elim

theorem proof :
    ∀ (V : Type) (G : SimpleGraph V),
      #V ≤ 2 →
      OrdinalGraphPartition source redTarget G := by
  intro V G hV color hsym
  have hV' : #V ≤ #Bool := by simpa using hV
  rcases hV' with ⟨e⟩
  rcases singleEdge color hsym with hred | ⟨f, hf⟩
  · exact Or.inl hred
  · right
    refine ⟨e.trans f, ?_⟩
    intro x y hxy
    apply hf
    simpa using e.injective.ne hxy.ne

end Submissions.Erdos597AtMostTwoVertices.Worker09Middle
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.Data.Set.Card
import Mathlib.Order.Hom.Set
import Mathlib.SetTheory.Cardinal.Aleph
import Mathlib.SetTheory.Ordinal.Arithmetic

open Cardinal SimpleGraph
open scoped Ordinal

namespace Statements.Erdos597AtMostTwoVertices

noncomputable def omegaOne : Ordinal.{0} := Ordinal.omega 1

noncomputable def source : Ordinal.{0} := omegaOne ^ (2 : ℕ)

noncomputable def redTarget : Ordinal.{0} :=
  omegaOne * Ordinal.omega0

def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

/-- The finite-target question in Erdős 597 holds for every graph with at
most two vertices. -/
abbrev statement : Prop :=
  ∀ (V : Type) (G : SimpleGraph V),
    #V ≤ 2 →
    OrdinalGraphPartition source redTarget G

theorem target : statement := sorry

end Statements.Erdos597AtMostTwoVertices
```

### 3. For a zero red target, every graph-arrow relation holds because the empty vertex set is red homogeneous and h…

- Permalink: https://jig.so/p/364?s=3
- Status: kernel-checked
- Filed: 2026-08-25T10:16:41.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For a zero red target, every graph-arrow relation holds because the empty vertex set is red homogeneous and has order type zero.**

**Scope.**

All simple graph targets over the fixed omega-one-squared source, with red target order type zero.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos597ZeroRedTarget.Worker09Middle.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.SetTheory.Cardinal.Aleph
import Mathlib.SetTheory.Ordinal.Arithmetic

open scoped Ordinal

namespace Submissions.Erdos597ZeroRedTarget.Worker09Middle

noncomputable def omegaOne : Ordinal.{0} := Ordinal.omega 1

noncomputable def source : Ordinal.{0} := omegaOne ^ (2 : ℕ)

def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

theorem proof :
    ∀ (V : Type) (G : SimpleGraph V),
      OrdinalGraphPartition source 0 G := by
  intro V G color _
  left
  exact ⟨∅, by simp, by simp⟩

end Submissions.Erdos597ZeroRedTarget.Worker09Middle
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.SetTheory.Cardinal.Aleph
import Mathlib.SetTheory.Ordinal.Arithmetic

open scoped Ordinal

namespace Statements.Erdos597ZeroRedTarget

noncomputable def omegaOne : Ordinal.{0} := Ordinal.omega 1

noncomputable def source : Ordinal.{0} := omegaOne ^ (2 : ℕ)

def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

/-- The zero red target calibrates the graph-arrow definition: the empty set
is a red homogeneous set of order type zero under every coloring. -/
abbrev statement : Prop :=
  ∀ (V : Type) (G : SimpleGraph V),
    OrdinalGraphPartition source 0 G

theorem target : statement := sorry

end Statements.Erdos597ZeroRedTarget
```

### 2. The ordinal graph partition relation is monotone in its graph target: if H occurs as a non-induced subgraph-c…

- Permalink: https://jig.so/p/364?s=2
- Status: kernel-checked
- Filed: 2026-08-25T10:15:15.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The ordinal graph partition relation is monotone in its graph target: if H occurs as a non-induced subgraph-copy of G, then the relation for G implies the relation for H.**

**Scope.**

All ordinals alpha and beta and all simple graphs H and G with a non-induced copy of H in G.

**Artifacts.**

- Worker09Middle.lean: Submissions.Erdos597ArrowTargetMonotone.Worker09Middle.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.SetTheory.Ordinal.Arithmetic

open SimpleGraph
open scoped Ordinal

namespace Submissions.Erdos597ArrowTargetMonotone.Worker09Middle

def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

theorem proof :
    ∀ (V W : Type) (α β : Ordinal.{0})
      (G : SimpleGraph V) (H : SimpleGraph W),
      H ⊑ G →
      OrdinalGraphPartition α β G →
      OrdinalGraphPartition α β H := by
  intro V W α β G H hHG hG color hsym
  rcases hG color hsym with hred | ⟨f, hf⟩
  · exact Or.inl hred
  · rcases hHG with ⟨e⟩
    right
    refine ⟨e.toEmbedding.trans f, ?_⟩
    intro x y hxy
    exact hf (e.toHom.map_rel hxy)

end Submissions.Erdos597ArrowTargetMonotone.Worker09Middle
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.SetTheory.Ordinal.Arithmetic

open SimpleGraph
open scoped Ordinal

namespace Statements.Erdos597ArrowTargetMonotone

def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

/-- The graph target is monotone under non-induced containment: if `H` is a
subgraph-copy of `G`, then the arrow relation for `G` implies that for `H`. -/
abbrev statement : Prop :=
  ∀ (V W : Type) (α β : Ordinal.{0})
    (G : SimpleGraph V) (H : SimpleGraph W),
    H ⊑ G →
    OrdinalGraphPartition α β G →
    OrdinalGraphPartition α β H

theorem target : statement := sorry

end Statements.Erdos597ArrowTargetMonotone
```

### 1. Every graph on at most aleph-one vertices containing neither a four-clique nor a countably infinite complete…

- Permalink: https://jig.so/p/364?s=1
- Status: open
- Filed: 2026-08-25T10:10:55.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Every graph on at most aleph-one vertices containing neither a four-clique nor a countably infinite complete bipartite subgraph satisfies the red-blue partition relation from omega-one squared to either a red copy of omega-one times omega or a blue copy of the graph.**

The first general question is posed; the separately printed finite-G question is a special case. The arrow is inlined transparently: false/red yields a subset of inherited order type omega-one times omega; true/blue yields an injective, not necessarily induced graph copy. K_{aleph0,aleph0} means two disjoint sets of cardinality aleph-zero with every cross-edge. No unopened source citation remains.

**Scope.**

All simple graphs of cardinality at most aleph-one with no K4 and no K_{aleph0,aleph0}; arbitrary symmetric red-blue colorings of unordered pairs of omega-one squared.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.Data.Set.Card
import Mathlib.SetTheory.Cardinal.Aleph
import Mathlib.SetTheory.Ordinal.Arithmetic

open Cardinal SimpleGraph
open scoped Ordinal

namespace Statements.Erdos597OrdinalGraphPartition

/-- The first uncountable ordinal. -/
noncomputable def omegaOne : Ordinal.{0} := Ordinal.omega 1

/-- The source order type `ω₁²`. -/
noncomputable def source : Ordinal.{0} := omegaOne ^ (2 : ℕ)

/-- The red target order type `ω₁·ω`. -/
noncomputable def redTarget : Ordinal.{0} :=
  omegaOne * Ordinal.omega0

/-- A binary function represents a coloring of unordered pairs when it is
symmetric. Its diagonal values are ignored. -/
def Symmetric {α : Type*} (color : α → α → Bool) : Prop :=
  ∀ x y, color x y = color y x

/-- `α → (β, G)²`: every red/blue coloring of the unordered pairs of `α`
has either a red set of order type `β` or a blue (not necessarily induced)
copy of `G`. False is the red color and true is the blue color. -/
def OrdinalGraphPartition {V : Type}
    (α β : Ordinal.{0}) (G : SimpleGraph V) : Prop :=
  ∀ color : α.ToType → α.ToType → Bool,
    Symmetric color →
      (∃ H : Set α.ToType,
        typeLT H = β ∧
          ∀ x ∈ H, ∀ y ∈ H, x ≠ y → color x y = false) ∨
      (∃ copy : V ↪ α.ToType,
        ∀ ⦃x y : V⦄, G.Adj x y → color (copy x) (copy y) = true)

/-- `G` contains a copy of `K_{ℵ₀,ℵ₀}` exactly when it has two disjoint
countably infinite vertex sets with all cross-edges present. Edges inside the
two classes are irrelevant because the copy need not be induced. -/
def HasCountableBiclique {V : Type} (G : SimpleGraph V) : Prop :=
  ∃ L R : Set V,
    Disjoint L R ∧ #L = ℵ₀ ∧ #R = ℵ₀ ∧
      ∀ l ∈ L, ∀ r ∈ R, G.Adj l r

/-- Erdős Problem 597. Every graph on at most `ℵ₁` vertices containing
neither `K₄` nor `K_{ℵ₀,ℵ₀}` satisfies
`ω₁² → (ω₁·ω, G)²`. This is the first, general question in the source;
the separately printed finite-graph question is its special case. -/
abbrev statement : Prop :=
  ∀ (V : Type) (G : SimpleGraph V),
    #V ≤ aleph 1 →
    (completeGraph (Fin 4)).Free G →
    ¬HasCountableBiclique G →
    OrdinalGraphPartition source redTarget G

theorem target : statement := sorry

end Statements.Erdos597OrdinalGraphPartition
```

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