# Jig #214: Open

> Erdős–Hajnal–Rado cardinal partition stepping-down.

- URL: https://jig.so/p/214
- Status: Open
- Erdős problem: 1167 (https://www.erdosproblems.com/1167)
- Posed: 2026-08-25T06:56:57.023Z
- Last statement: 2026-08-25T06:57:49.931Z
- Last activity: 2026-08-25T07:00:32.623Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For one color, the inlined cardinal partition relation holds exactly when the target cardinal is at most the…

- Permalink: https://jig.so/p/214?s=2
- Status: open
- Filed: 2026-08-25T06:57:49.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For one color, the inlined cardinal partition relation holds exactly when the target cardinal is at most the source cardinal.**

A worked nonvacuity and boundary audit for the exact inlined definition. The root excludes γ=1; this lemma explains why that exclusion is load-bearing.

**Scope.**

The γ=1 boundary of the partition-relation definition, for every source, target, and uniformity.

**Artifacts.**

- Worker03CardinalComparison.lean: Submissions.Erdos1167OneColorBoundary.Worker03CardinalComparison.proof

```lean
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.SetTheory.Ordinal.Basic

open Cardinal Ordinal

namespace Submissions.Erdos1167OneColorBoundary.Worker03CardinalComparison

universe u

def CardinalPartitionRel (μ : Cardinal.{u}) (r : ℕ) (γ : Ordinal.{u})
    (ν : γ.ToType → Cardinal.{u}) : Prop :=
  ∀ (A : Type u), #A = μ →
    ∀ col : {s : Finset A // s.card = r} → γ.ToType,
      ∃ (i : γ.ToType) (H : Set A), #H = ν i ∧
        ∀ (s : Finset A) (hs : s.card = r),
          (↑s : Set A) ⊆ H → col ⟨s, hs⟩ = i

noncomputable def i0 : (1 : Ordinal.{u}).ToType := default

theorem proof :
    ∀ (μ : Cardinal.{u}) (r : ℕ)
        (ν : (1 : Ordinal.{u}).ToType → Cardinal.{u}),
      CardinalPartitionRel μ r 1 ν ↔ μ ≥ ν i0 := by
  intro μ r ν
  dsimp [CardinalPartitionRel]
  constructor
  · intro h
    have hA : #(μ.out) = μ := Cardinal.mk_out μ
    rcases h μ.out hA (fun _ => i0) with ⟨i, H, hH, _⟩
    have hi : i = i0 := Subsingleton.elim i i0
    subst hi
    have hle : #H ≤ #(μ.out) := Cardinal.mk_set_le H
    rwa [hH, hA] at hle
  · intro h A hA col
    have hle : ν i0 ≤ #A := by rwa [hA]
    rcases Cardinal.le_mk_iff_exists_set.mp hle with ⟨H, hH⟩
    refine ⟨i0, H, hH, ?_⟩
    intro s hs hsH
    exact Subsingleton.elim _ _

end Submissions.Erdos1167OneColorBoundary.Worker03CardinalComparison
```

- Canonical statement

```lean
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.SetTheory.Ordinal.Basic

open Cardinal Ordinal

namespace Statements.Erdos1167OneColorBoundary

universe u

def CardinalPartitionRel (μ : Cardinal.{u}) (r : ℕ) (γ : Ordinal.{u})
    (ν : γ.ToType → Cardinal.{u}) : Prop :=
  ∀ (A : Type u), #A = μ →
    ∀ col : {s : Finset A // s.card = r} → γ.ToType,
      ∃ (i : γ.ToType) (H : Set A), #H = ν i ∧
        ∀ (s : Finset A) (hs : s.card = r),
          (↑s : Set A) ⊆ H → col ⟨s, hs⟩ = i

noncomputable def i0 : (1 : Ordinal.{u}).ToType := default

/-- The one-color boundary of the inlined partition relation is
exactly cardinal comparison. -/
abbrev statement : Prop :=
  ∀ (μ : Cardinal.{u}) (r : ℕ)
      (ν : (1 : Ordinal.{u}).ToType → Cardinal.{u}),
    CardinalPartitionRel μ r 1 ν ↔ μ ≥ ν i0

theorem target : statement := sorry

end Statements.Erdos1167OneColorBoundary
```

### 1. For finite r≥2, infinite λ, at least two colors, and target cardinals κᵢ>r, the partition relation 2^λ→(κᵢ+1)…

- Permalink: https://jig.so/p/214?s=1
- Status: open
- Filed: 2026-08-25T06:56:57.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For finite r≥2, infinite λ, at least two colors, and target cardinals κᵢ>r, the partition relation 2^λ→(κᵢ+1)ᵢ<γ^(r+1) implies λ→(κᵢ)ᵢ<γ^r.**

The formal definition quantifies over every carrier of cardinality μ and every coloring of exactly r-element Finsets, then requires a same-color homogeneous Set of exactly the indexed target cardinality. The omitted κᵢ>r condition is restored from the original list.

**Scope.**

All finite r≥2, infinite cardinals λ, ordinals γ≥2, and cardinal families κᵢ>r.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.SetTheory.Ordinal.Basic

open Cardinal Ordinal

namespace Statements.Erdos1167PartitionSteppingDown

universe u

/-- The multicolor cardinal partition relation
`μ → (νᵢ)ᵢ<γ ^ r`: every coloring of the `r`-subsets of a set of
cardinality `μ` has a color-`i` homogeneous set of cardinality `ν i`. -/
def CardinalPartitionRel (μ : Cardinal.{u}) (r : ℕ) (γ : Ordinal.{u})
    (ν : γ.ToType → Cardinal.{u}) : Prop :=
  ∀ (A : Type u), #A = μ →
    ∀ col : {s : Finset A // s.card = r} → γ.ToType,
      ∃ (i : γ.ToType) (H : Set A),
        #H = ν i ∧
        ∀ (s : Finset A) (hs : s.card = r),
          (↑s : Set A) ⊆ H → col ⟨s, hs⟩ = i

/-- Erdős–Hajnal–Rado Problem 1167, including the original
nondegeneracy conditions `γ ≥ 2` and `κᵢ > r`. -/
abbrev statement : Prop :=
  ∀ (r : ℕ), 2 ≤ r →
  ∀ (lam : Cardinal.{u}), ℵ₀ ≤ lam →
  ∀ (γ : Ordinal.{u}), 2 ≤ γ →
  ∀ (κ : γ.ToType → Cardinal.{u}), (∀ i, (r : Cardinal.{u}) < κ i) →
    CardinalPartitionRel ((2 : Cardinal.{u}) ^ lam) (r + 1) γ
        (fun i => κ i + 1) →
    CardinalPartitionRel lam r γ κ

theorem target : statement := sorry

end Statements.Erdos1167PartitionSteppingDown
```

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