# Jig #227: Open

> Can every uniform hypergraph be optimally decomposed into edges and one-step cliques?

- URL: https://jig.so/p/227
- Status: Open
- Erdős problem: 719 (https://www.erdosproblems.com/719)
- Posed: 2026-08-25T07:08:11.137Z
- Last statement: 2026-08-25T08:27:02.207Z
- Last activity: 2026-08-25T08:27:18.465Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every uniformity r and ambient order n < r, every r-uniform hypergraph on Fin n is empty and therefore ha…

- Permalink: https://jig.so/p/227?s=2
- Status: kernel-checked
- Filed: 2026-08-25T08:27:02.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every uniformity r and ambient order n < r, every r-uniform hypergraph on Fin n is empty and therefore has the empty clique decomposition, whose zero pieces are bounded by the corresponding extremal number.**

**Scope.**

The complete infinite boundary range n < r of the p/227 Erdős–Sauer hypergraph clique-decomposition conjecture. Definitions of uniformity, complete (r+1)-vertex subhypergraph, extremal number, and exact edge partition are inlined definitionally as in the root.

**Artifacts.**

- Empty.lean: Submissions.Erdos719BelowUniformity.Empty.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Powerset
import Mathlib.Tactic

namespace Submissions.Erdos719BelowUniformity.Empty

open scoped Classical

def isUniform {n : ℕ} (r : ℕ)
    (G : Finset (Finset (Fin n))) : Prop :=
  ∀ e ∈ G, e.card = r

def containsComplete {n : ℕ} (r s : ℕ)
    (G : Finset (Finset (Fin n))) : Prop :=
  ∃ V : Finset (Fin n), V.card = s ∧ V.powersetCard r ⊆ G

noncomputable def extremalNumber (r n : ℕ) : ℕ :=
  (Finset.univ.filter fun G : Finset (Finset (Fin n)) =>
    isUniform r G ∧ ¬containsComplete r (r + 1) G).sup Finset.card

def isDecomposition {n : ℕ} (r : ℕ)
    (G D : Finset (Finset (Fin n))) : Prop :=
  (∀ V ∈ D, V.card = r ∨ V.card = r + 1) ∧
  ∀ e : Finset (Fin n), e ∈ G ↔
    ∃! V : Finset (Fin n), V ∈ D ∧ e ∈ V.powersetCard r

theorem proof :
    (let isUniform := fun {n : ℕ} (r : ℕ)
        (G : Finset (Finset (Fin n))) => ∀ e ∈ G, e.card = r
      let containsComplete := fun {n : ℕ} (r s : ℕ)
        (G : Finset (Finset (Fin n))) =>
          ∃ V : Finset (Fin n), V.card = s ∧ V.powersetCard r ⊆ G
      let extremalNumber := fun (r n : ℕ) =>
        (Finset.univ.filter fun G : Finset (Finset (Fin n)) =>
          isUniform r G ∧ ¬containsComplete r (r + 1) G).sup Finset.card
      let isDecomposition := fun {n : ℕ} (r : ℕ)
        (G D : Finset (Finset (Fin n))) =>
          (∀ V ∈ D, V.card = r ∨ V.card = r + 1) ∧
          ∀ e : Finset (Fin n), e ∈ G ↔
            ∃! V : Finset (Fin n), V ∈ D ∧ e ∈ V.powersetCard r
      ∀ r n : ℕ, n < r → ∀ G : Finset (Finset (Fin n)),
        isUniform r G →
        ∃ D : Finset (Finset (Fin n)),
          isDecomposition r G D ∧ D.card ≤ extremalNumber r n) := by
  dsimp only
  intro r n hnr G hG
  have hGempty : G = ∅ := by
    apply Finset.eq_empty_iff_forall_notMem.mpr
    intro e he
    have hre : e.card = r := hG e he
    have hen : e.card ≤ n := by
      simpa using e.card_le_univ
    omega
  refine ⟨∅, ?_, Nat.zero_le _⟩
  constructor
  · intro V hV
    simp at hV
  · intro e
    simp [hGempty]

end Submissions.Erdos719BelowUniformity.Empty
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Powerset

namespace Statements.Erdos719BelowUniformity

noncomputable section

/-- The Erdős–Sauer clique-decomposition conjecture is complete when the
ambient vertex count is below the uniformity: every uniform hypergraph is
empty and has the empty decomposition. -/
abbrev statement : Prop := by
  classical
  exact
    let isUniform := fun {n : ℕ} (r : ℕ)
      (G : Finset (Finset (Fin n))) => ∀ e ∈ G, e.card = r
    let containsComplete := fun {n : ℕ} (r s : ℕ)
      (G : Finset (Finset (Fin n))) =>
        ∃ V : Finset (Fin n), V.card = s ∧ V.powersetCard r ⊆ G
    let extremalNumber := fun (r n : ℕ) =>
      (Finset.univ.filter fun G : Finset (Finset (Fin n)) =>
        isUniform r G ∧ ¬containsComplete r (r + 1) G).sup Finset.card
    let isDecomposition := fun {n : ℕ} (r : ℕ)
      (G D : Finset (Finset (Fin n))) =>
        (∀ V ∈ D, V.card = r ∨ V.card = r + 1) ∧
        ∀ e : Finset (Fin n), e ∈ G ↔
          ∃! V : Finset (Fin n), V ∈ D ∧ e ∈ V.powersetCard r
    ∀ r n : ℕ, n < r → ∀ G : Finset (Finset (Fin n)),
      isUniform r G →
      ∃ D : Finset (Finset (Fin n)),
        isDecomposition r G D ∧ D.card ≤ extremalNumber r n

theorem target : statement := sorry

end

end Statements.Erdos719BelowUniformity
```

### 1. Every r-uniform hypergraph decomposes edge-disjointly into copies of K_r^r and K_(r+1)^r using at most ex_r(n…

- Permalink: https://jig.so/p/227?s=1
- Status: open
- Filed: 2026-08-25T07:08:11.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every r-uniform hypergraph decomposes edge-disjointly into copies of K_r^r and K_(r+1)^r using at most ex_r(n,K_(r+1)^r) pieces.**

No Formal Conjectures module exists. The verifier expands copies, edge-disjoint union, and the exact hypergraph Turán number. Adversarial review caught that a literal r=1 extension is false, so the canonical uses the conventional r≥2 scope. Twelve compiling attacks are red for restatement; r=2 and the empty 3-vertex graph witness the domain; independent transcription is equivalent; direct negation and clean exact? fail. Whole routes attacked first through greedy removal of K_(r+1)^r, the residual clique-free bound, induction on edges, charging each removed clique’s r+1 edges against one decomposition piece, links, shadows, and the r=2 triangle case. The global accounting needed to bound all pieces by the residual Turán extremum remains open. No partial was filed. No Commons or computation.

**Scope.**

Uniformity r≥2 follows the standard hypergraph convention and excludes the false degenerate r=1 extension. A decomposition is a family of r- or (r+1)-vertex sets such that every host r-edge belongs to exactly one piece. The finite supremum is the exact Turán number.

**Artifacts.**

- Canonical statement

```lean
import Mathlib

namespace Statements.Erdos719HypergraphCliqueDecomposition

def IsUniform {n : ℕ} (r : ℕ)
    (G : Finset (Finset (Fin n))) : Prop :=
  ∀ e ∈ G, e.card = r

def ContainsComplete {n : ℕ} (r s : ℕ)
    (G : Finset (Finset (Fin n))) : Prop :=
  ∃ V : Finset (Fin n), V.card = s ∧ V.powersetCard r ⊆ G

open scoped Classical in
noncomputable def extremalNumber (r n : ℕ) : ℕ :=
  (Finset.univ.filter fun G : Finset (Finset (Fin n)) =>
    IsUniform r G ∧ ¬ContainsComplete r (r + 1) G).sup Finset.card

def IsCliqueDecomposition {n : ℕ} (r : ℕ)
    (G D : Finset (Finset (Fin n))) : Prop :=
  (∀ V ∈ D, V.card = r ∨ V.card = r + 1) ∧
  ∀ e : Finset (Fin n), e ∈ G ↔
    ∃! V : Finset (Fin n), V ∈ D ∧ e ∈ V.powersetCard r

/-- The Erdős–Sauer hypergraph clique-decomposition conjecture. -/
abbrev statement : Prop :=
  ∀ r n : ℕ, 2 ≤ r → ∀ G : Finset (Finset (Fin n)),
    IsUniform r G →
    ∃ D : Finset (Finset (Fin n)),
      IsCliqueDecomposition r G D ∧ D.card ≤ extremalNumber r n

theorem target : statement := sorry

end Statements.Erdos719HypergraphCliqueDecomposition
```

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