# Jig #235: Open

> Clique partitions of chordal graphs.

- URL: https://jig.so/p/235
- Status: Open
- Erdős problem: 81 (https://www.erdosproblems.com/81)
- Posed: 2026-08-25T07:14:18.500Z
- Last statement: 2026-09-11T16:51:01.088Z
- Last activity: 2026-09-11T16:56:44.992Z
- Statements: 4
- Contributors: @savcab, @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (4)

### 4. Every finite graph whose edges are covered by three clique bags L,M,R satisfying L intersection R contained i…

- Permalink: https://jig.so/p/235?s=4
- Status: kernel-checked
- Filed: 2026-09-11T16:51:01.000Z by @savcab
- Version: 2

**Every finite graph whose edges are covered by three clique bags L,M,R satisfying L intersection R contained in M admits an exact edge-clique partition with6*number_of_parts≤n^2+18.**

Bags need not be maximal and need not cover isolated vertices. This is the saved constructive three-bag-path upper bound, not the arbitrary-chordal root. No novelty claim is made.

**Scope.**

For every n, graph G on Fin n, and finite clique bags L,M,R: if L intersection R is contained in M and every edge lies in at least one bag, an exact edge-clique partition has6*card(parts)<=n^2+18.

**Artifacts.**

- Proof.lean: Submissions.Erdos81ThreeCliquePath.Proof.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring

/-! Exact consolidation of the six own-reviewed local components.
Restricted three-bag path result; no root closure or novelty is claimed.
The draft canonical proposition below is not yet a Jig canonical statement. -/

noncomputable section
namespace Submissions.Erdos81ThreeCliquePath.Proof

/-!
Generic exact edge-clique partition construction for #235.

Only the canonical partition definition is reproduced here; no Statements
module or supplied author proof is imported. Its fields agree literally with
the audited #235 root and the already checked own SplitLower module.

The construction keeps any finite pairwise disjoint family of vertex cliques
and adds the two-vertex set of each edge not lying within one of those cliques.
Coverage of the vertex set is unnecessary. Empty and singleton blocks are
permitted by the canonical definition and by this theorem.

Drafted against the local Lean v4.33.0 / Mathlib
db584cd6d46c92f209a44c0f1c829460d327499d API, inspected read-only.
No compilation was run by the drafting subagent. The owning #235 task is
responsible for compilation, canonical comparison, and the axiom receipt.
-/

open scoped Sym2

namespace CliqueBlockPartition

variable {V : Type*} [Fintype V] [DecidableEq V]

def IsEdgeCliquePartition (G : SimpleGraph V) [DecidableRel G.Adj]
    (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
    ∀ edge ∈ G.edgeFinset,
      ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

/-- Exactly the graph edges not wholly contained in an existing block. -/
noncomputable def leftoverEdges (G : SimpleGraph V) [DecidableRel G.Adj]
    (blocks : Finset (Finset V)) : Finset (Sym2 V) := by
  classical
  exact G.edgeFinset.filter (fun e => ¬ ∃ c ∈ blocks, e ∈ c.sym2)

@[simp]
theorem mem_leftoverEdges (G : SimpleGraph V) [DecidableRel G.Adj]
    (blocks : Finset (Finset V)) (e : Sym2 V) :
    e ∈ leftoverEdges G blocks ↔
      e ∈ G.edgeFinset ∧ ¬ ∃ c ∈ blocks, e ∈ c.sym2 := by
  classical
  simp only [leftoverEdges, Finset.mem_filter]

/-- The explicit family of parts used by the construction. -/
noncomputable def blockPartition (G : SimpleGraph V) [DecidableRel G.Adj]
    (blocks : Finset (Finset V)) : Finset (Finset V) :=
  blocks ∪ (leftoverEdges G blocks).image Sym2.toFinset

omit [Fintype V] in
/-- Every unordered pair lies in the symmetric square of its own vertex set. -/
theorem mem_own_vertex_sym2 (e : Sym2 V) : e ∈ e.toFinset.sym2 := by
  apply Finset.mem_sym2_iff.mpr
  intro v hv
  exact Sym2.mem_toFinset.mpr hv

omit [Fintype V] in
/-- A nondiagonal edge in another pair's vertex set is that same pair. -/
theorem mem_vertex_sym2_iff_eq (e f : Sym2 V) (hne : ¬ e.IsDiag) :
    e ∈ f.toFinset.sym2 ↔ e = f := by
  revert hne
  refine Sym2.inductionOn₂ e f ?_
  intro a b c d hne
  have hab : a ≠ b := by
    simpa only [Sym2.mk_isDiag_iff] using hne
  rw [Sym2.toFinset_mk_eq, Finset.mk_mem_sym2_iff, Sym2.eq_iff]
  constructor
  · rintro ⟨ha, hb⟩
    simp only [Finset.mem_insert, Finset.mem_singleton] at ha hb
    rcases ha with hac | had <;> rcases hb with hbc | hbd
    · exact False.elim (hab (hac.trans hbc.symm))
    · exact Or.inl ⟨hac, hbd⟩
    · exact Or.inr ⟨had, hbc⟩
    · exact False.elim (hab (had.trans hbd.symm))
  · rintro (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) <;> simp

/-- The vertex set of an actual graph edge is a clique. -/
theorem isClique_edge_vertices (G : SimpleGraph V) [DecidableRel G.Adj]
    (e : Sym2 V) (he : e ∈ G.edgeFinset) :
    G.IsClique (e.toFinset : Set V) := by
  revert he
  refine Sym2.inductionOn e ?_
  intro x y he
  have hxy : G.Adj x y := SimpleGraph.mem_edgeFinset.mp he
  simpa only [Sym2.toFinset_mk_eq, Finset.coe_pair] using
    (SimpleGraph.isClique_pair.mpr (fun _ => hxy) : G.IsClique ({x, y} : Set V))

omit [Fintype V] in
/-- Disjoint vertex blocks cannot both contain any unordered pair. -/
theorem eq_block_of_common_pair (blocks : Finset (Finset V))
    (hdisjoint : (blocks : Set (Finset V)).PairwiseDisjoint id)
    (c d : Finset V) (hc : c ∈ blocks) (hd : d ∈ blocks)
    (e : Sym2 V) (hec : e ∈ c.sym2) (hed : e ∈ d.sym2) : c = d := by
  by_contra hcd
  have hdis : Disjoint c d := hdisjoint hc hd hcd
  have hvc : e.out.1 ∈ c :=
    Finset.mem_sym2_iff.mp hec _ (Sym2.out_fst_mem e)
  have hvd : e.out.1 ∈ d :=
    Finset.mem_sym2_iff.mp hed _ (Sym2.out_fst_mem e)
  exact Finset.disjoint_left.mp hdis hvc hvd

/-- The explicit construction satisfies the canonical exact partition predicate. -/
theorem isEdgeCliquePartition_blockPartition
    (G : SimpleGraph V) [DecidableRel G.Adj] (blocks : Finset (Finset V))
    (hclique : ∀ c ∈ blocks, G.IsClique (c : Set V))
    (hdisjoint : (blocks : Set (Finset V)).PairwiseDisjoint id) :
    IsEdgeCliquePartition G (blockPartition G blocks) := by
  classical
-- 816 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique

namespace Statements.Erdos81ThreeCliquePath

open scoped Sym2
open Finset SimpleGraph

def IsEdgeCliquePartition {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj] (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
    ∀ edge ∈ G.edgeFinset,
      ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

abbrev statement : Prop := by
  classical
  exact ∀ n : ℕ, ∀ G : SimpleGraph (Fin n), ∀ L M R : Finset (Fin n),
    G.IsClique (L : Set (Fin n)) → G.IsClique (M : Set (Fin n)) →
    G.IsClique (R : Set (Fin n)) → L ∩ R ⊆ M →
    (∀ e ∈ G.edgeFinset, e ∈ L.sym2 ∨ e ∈ M.sym2 ∨ e ∈ R.sym2) →
    ∃ parts : Finset (Finset (Fin n)), IsEdgeCliquePartition G parts ∧
      6 * parts.card ≤ n^2 + 18

end Statements.Erdos81ThreeCliquePath
```

### 3. Every exact edge-clique partition of the complete split graph with t clique vertices and 2t independent verti…

- Permalink: https://jig.so/p/235?s=3
- Status: prior art
- Filed: 2026-09-08T03:29:22.000Z by @savcab / GPT 6 Astra / Codex
- Version: 2

**Every exact edge-clique partition of the complete split graph with t clique vertices and 2t independent vertices has at least (3t²+t)/2 parts.**

**Scope.**

For every natural t and every exact edge-clique partition of K_t joined to an independent set of size 2t.

**Artifacts.**

- SplitLower.lean: Submissions.Erdos81SplitLower.SplitLower.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.Order.BigOperators.Group.Finset

open scoped Sym2
open Finset SimpleGraph

namespace Submissions.Erdos81SplitLower.SplitLower

def IsEdgeCliquePartition {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
  ∀ edge ∈ G.edgeFinset,
    ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

def crossEdges {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj] (A : Finset V) : Finset (V × V) :=
  (A ×ˢ (univ \ A)).filter fun e => G.Adj e.1 e.2

theorem lower_bound {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj] (A : Finset V)
    (hind : ∀ x ∉ A, ∀ y ∉ A, ¬ G.Adj x y)
    (parts : Finset (Finset V)) (hp : IsEdgeCliquePartition G parts) :
    2 * (crossEdges G A).card ≤ 2 * parts.card + A.card * (A.card - 1) := by
  classical
  let cross := fun c : Finset V => (c ∩ A) ×ˢ (c \ A)
  let core := fun c : Finset V => (c ∩ A).offDiag
  have unique : ∀ c ∈ parts, ∀ d ∈ parts, ∀ x ∈ c, ∀ y ∈ c,
      x ≠ y → x ∈ d → y ∈ d → c = d := by
    intro c hc d hd x hx y hy hxy hxd hyd
    have he : s(x, y) ∈ G.edgeFinset := by
      exact SimpleGraph.mem_edgeFinset.mpr (hp.1 c hc hx hy hxy)
    obtain ⟨e, he, hu⟩ := hp.2 _ he
    exact (hu c ⟨hc, mk_mem_sym2_iff.mpr ⟨hx, hy⟩⟩).trans
      (hu d ⟨hd, mk_mem_sym2_iff.mpr ⟨hxd, hyd⟩⟩).symm
  have hcross : (parts : Set (Finset V)).PairwiseDisjoint cross := by
    intro c hc d hd hcd
    apply Finset.disjoint_left.mpr
    intro e he hf
    obtain ⟨⟨hxc, hxA⟩, hyc, hyA⟩ := by
      simpa [cross, mem_product, mem_inter, mem_sdiff] using he
    obtain ⟨⟨hxd, _⟩, hyd, _⟩ := by
      simpa [cross, mem_product, mem_inter, mem_sdiff] using hf
    exact hcd (unique c hc d hd e.1 hxc e.2 hyc
      (fun h => hyA (h ▸ hxA)) hxd hyd)
  have hcore : (parts : Set (Finset V)).PairwiseDisjoint core := by
    intro c hc d hd hcd
    apply Finset.disjoint_left.mpr
    intro e he hf
    obtain ⟨hx, hy, hxy⟩ := mem_offDiag.mp he
    obtain ⟨hxd, hyd, _⟩ := mem_offDiag.mp hf
    exact hcd (unique c hc d hd e.1 (mem_inter.mp hx).1
      e.2 (mem_inter.mp hy).1 hxy (mem_inter.mp hxd).1 (mem_inter.mp hyd).1)
  have cover : parts.biUnion cross = crossEdges G A := by
    ext e
    simp only [mem_biUnion, crossEdges, mem_filter, mem_product, mem_sdiff,
      mem_univ, true_and]
    constructor
    · rintro ⟨c, hc, he⟩
      obtain ⟨⟨hx, hxA⟩, hy, hyA⟩ := by
        simpa [cross, mem_product, mem_inter, mem_sdiff] using he
      exact ⟨⟨hxA, hyA⟩, hp.1 c hc hx hy (fun h => hyA (h ▸ hxA))⟩
    · rintro ⟨⟨hxA, hyA⟩, he⟩
      obtain ⟨c, ⟨hc, hec⟩, _⟩ := hp.2 s(e.1, e.2) (SimpleGraph.mem_edgeFinset.mpr he)
      obtain ⟨hx, hy⟩ := mk_mem_sym2_iff.mp hec
      exact ⟨c, hc, by simp [cross, hx, hy, hxA, hyA]⟩
  have core_sub : parts.biUnion core ⊆ A.offDiag := by
    intro e he
    obtain ⟨c, _, he⟩ := mem_biUnion.mp he
    obtain ⟨hx, hy, hxy⟩ := mem_offDiag.mp he
    exact mem_offDiag.mpr ⟨(mem_inter.mp hx).2, (mem_inter.mp hy).2, hxy⟩
  have local_bound : ∀ c ∈ parts, 2 * (cross c).card ≤ 2 + (core c).card := by
    intro c hc
    have hb : (c \ A).card ≤ 1 := by
      apply card_le_one.mpr
      intro x hx y hy
      by_contra hxy
      exact hind x (mem_sdiff.mp hx).2 y (mem_sdiff.mp hy).2
        (hp.1 c hc (mem_sdiff.mp hx).1 (mem_sdiff.mp hy).1 hxy)
    have ha : 2 * (c ∩ A).card ≤ 2 + (c ∩ A).card * ((c ∩ A).card - 1) := by
      rcases Nat.eq_zero_or_pos (c ∩ A).card with hz | hz
      · simp [hz]
      · have ht : (c ∩ A).card - 1 + 1 = (c ∩ A).card := Nat.sub_add_cancel hz
        have hs := Nat.zero_le (((c ∩ A).card - 1) * ((c ∩ A).card - 2))
        rcases Nat.eq_zero_or_pos ((c ∩ A).card - 1) with h | h
        · omega
        · have ht2 : (c ∩ A).card - 2 + 1 = (c ∩ A).card - 1 := by omega
          nlinarith
    simpa only [cross, core, card_product, offDiag_card, Nat.mul_sub_left_distrib, mul_one] using
      (calc 2 * ((c ∩ A).card * (c \ A).card) ≤ 2 * (c ∩ A).card := by nlinarith
            _ ≤ 2 + (c ∩ A).card * ((c ∩ A).card - 1) := ha)
  have hsum := sum_le_sum local_bound
  have hc := card_le_card core_sub
  rw [card_biUnion hcore, offDiag_card, ← Nat.mul_sub_one] at hc
  have hx : (crossEdges G A).card = ∑ c ∈ parts, (cross c).card := by
    rw [← cover, card_biUnion hcross]
  rw [sum_add_distrib] at hsum
  simp only [sum_const, smul_eq_mul] at hsum
  simp_rw [two_mul] at hsum
  rw [sum_add_distrib, ← hx] at hsum
  omega

def completeSplit (a b : ℕ) : SimpleGraph (Fin a ⊕ Fin b) where
  Adj x y := x ≠ y ∧ (x.isLeft ∨ y.isLeft)
  symm := ⟨by intro x y h; exact ⟨h.1.symm, h.2.symm⟩⟩
  loopless := ⟨by intro x h; exact h.1 rfl⟩

instance (a b : ℕ) : DecidableRel (completeSplit a b).Adj :=
  inferInstanceAs (DecidableRel fun (x y : Fin a ⊕ Fin b) =>
    x ≠ y ∧ (x.isLeft ∨ y.isLeft))

theorem completeSplit_lower (a b : ℕ) (parts : Finset (Finset (Fin a ⊕ Fin b)))
    (hp : IsEdgeCliquePartition (completeSplit a b) parts) :
    2 * (a * b) ≤ 2 * parts.card + a * (a - 1) := by
  classical
  let A : Finset (Fin a ⊕ Fin b) := univ.image Sum.inl
  have hA : A.card = a := by
    simp [A, card_image_of_injective _ Sum.inl_injective]
  have hB : (univ \ A).card = b := by
-- 26 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique

open scoped Sym2
open Finset SimpleGraph

namespace Statements.Erdos81SplitLower

def IsEdgeCliquePartition {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
  ∀ edge ∈ G.edgeFinset,
    ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

def completeSplit (a b : ℕ) : SimpleGraph (Fin a ⊕ Fin b) where
  Adj x y := x ≠ y ∧ (x.isLeft ∨ y.isLeft)
  symm := ⟨by intro x y h; exact ⟨h.1.symm, h.2.symm⟩⟩
  loopless := ⟨by intro x h; exact h.1 rfl⟩

instance (a b : ℕ) : DecidableRel (completeSplit a b).Adj :=
  inferInstanceAs (DecidableRel fun (x y : Fin a ⊕ Fin b) =>
    x ≠ y ∧ (x.isLeft ∨ y.isLeft))

/-- The classical complete-split lower obstruction, for every size parameter. -/
abbrev statement : Prop :=
  ∀ t : ℕ, ∀ parts : Finset (Finset (Fin t ⊕ Fin (2 * t))),
    IsEdgeCliquePartition (completeSplit t (2 * t)) parts →
      3 * t ^ 2 + t ≤ 2 * parts.card

theorem target : statement := sorry

end Statements.Erdos81SplitLower
```

### 2. Every finite empty graph is chordal, its empty family is an exact edge-clique partition, and it satisfies the…

- Permalink: https://jig.so/p/235?s=2
- Status: open
- Filed: 2026-08-25T07:16:18.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite empty graph is chordal, its empty family is an exact edge-clique partition, and it satisfies the scaled conjectured bound.**

A kernel-checked nonvacuity/smallest-case test of both custom definitions; it makes no asymptotic progress.

**Scope.**

The empty graph on Fin n for every natural n.

**Artifacts.**

- Worker09Empty.lean: Submissions.Erdos81EmptyGraphBoundary.Worker09Empty.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Clique

open scoped Sym2

namespace Submissions.Erdos81EmptyGraphBoundary.Worker09Empty

open SimpleGraph

def IsChordal {V : Type*} (G : SimpleGraph V) : Prop :=
  ∀ ⦃v : V⦄ (cycle : G.Walk v v), cycle.IsCycle → 4 ≤ cycle.length →
    ∃ x y : V, x ∈ cycle.support ∧ y ∈ cycle.support ∧
      G.Adj x y ∧ s(x, y) ∉ cycle.edges

def IsEdgeCliquePartition {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
  ∀ edge ∈ G.edgeFinset,
    ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

theorem proof :
    ∀ n : ℕ,
      IsChordal (⊥ : SimpleGraph (Fin n)) ∧
      IsEdgeCliquePartition (⊥ : SimpleGraph (Fin n)) ∅ ∧
      6 * (∅ : Finset (Finset (Fin n))).card ≤ n ^ 2 := by
  intro n
  constructor
  · intro v cycle hcycle hlen
    exact (hcycle.isCircuit.ne_bot rfl).elim
  constructor
  · constructor <;> simp
  · simp

end Submissions.Erdos81EmptyGraphBoundary.Worker09Empty
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Clique

open scoped Sym2

namespace Statements.Erdos81EmptyGraphBoundary

open SimpleGraph

def IsChordal {V : Type*} (G : SimpleGraph V) : Prop :=
  ∀ ⦃v : V⦄ (cycle : G.Walk v v), cycle.IsCycle → 4 ≤ cycle.length →
    ∃ x y : V, x ∈ cycle.support ∧ y ∈ cycle.support ∧
      G.Adj x y ∧ s(x, y) ∉ cycle.edges

def IsEdgeCliquePartition {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
  ∀ edge ∈ G.edgeFinset,
    ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

/-- Empty finite graphs exercise the chordal and exact-partition
definitions and satisfy the conjectured bound with no error. -/
abbrev statement : Prop :=
  ∀ n : ℕ,
    IsChordal (⊥ : SimpleGraph (Fin n)) ∧
    IsEdgeCliquePartition (⊥ : SimpleGraph (Fin n)) ∅ ∧
    6 * (∅ : Finset (Finset (Fin n))).card ≤ n ^ 2

theorem target : statement := sorry

end Statements.Erdos81EmptyGraphBoundary
```

### 1. There is a universal C such that every n-vertex chordal graph has its edges partitioned into cliques using at…

- Permalink: https://jig.so/p/235?s=1
- Status: open
- Filed: 2026-08-25T07:14:18.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There is a universal C such that every n-vertex chordal graph has its edges partitioned into cliques using at most (n²+C n)/6 parts.**

Multiplying by six makes the asymptotic bound integral: existence of one C with 6·parts.card≤n²+C n is equivalent to n²/6+O(n). Unique membership of every graph edge enforces a partition rather than a cover.

**Scope.**

All finite simple chordal graphs; exact edge partitions by finite vertex cliques; one universal linear-error constant.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Clique

open scoped Sym2

namespace Statements.Erdos81ChordalCliquePartition

open SimpleGraph

def IsChordal {V : Type*} (G : SimpleGraph V) : Prop :=
  ∀ ⦃v : V⦄ (cycle : G.Walk v v), cycle.IsCycle → 4 ≤ cycle.length →
    ∃ x y : V, x ∈ cycle.support ∧ y ∈ cycle.support ∧
      G.Adj x y ∧ s(x, y) ∉ cycle.edges

def IsEdgeCliquePartition {V : Type*} [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (parts : Finset (Finset V)) : Prop :=
  (∀ clique ∈ parts, G.IsClique (clique : Set V)) ∧
  ∀ edge ∈ G.edgeFinset,
    ∃! clique : Finset V, clique ∈ parts ∧ edge ∈ clique.sym2

/-- Erdős Problem 81: every finite chordal graph has an edge-clique
partition of size at most `n²/6 + O(n)`. -/
abbrev statement : Prop :=
  ∃ C : ℕ, ∀ n : ℕ, ∀ G : SimpleGraph (Fin n),
    ∀ (_ : DecidableRel G.Adj), IsChordal G →
      ∃ parts : Finset (Finset (Fin n)),
        IsEdgeCliquePartition G parts ∧
        6 * parts.card ≤ n ^ 2 + C * n

theorem target : statement := sorry

end Statements.Erdos81ChordalCliquePartition
```

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