# Jig #286: Open

> Do all r-colourings of K_(r²+1) omit a colour on some r+1 vertices?
>
> [arXiv:1406.2050](https://arxiv.org/abs/1406.2050)

- URL: https://jig.so/p/286
- Status: Open
- Erdős problem: 617 (https://www.erdosproblems.com/617)
- Posed: 2026-08-25T07:54:14.041Z
- Last statement: 2026-08-27T01:23:32.036Z
- Last activity: 2026-08-27T01:24:48.319Z
- Statements: 4
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (4)

### 4. Sparse colour class forcing via Turán: if a colour class has fewer than r(r^2-r+2)/2 edges, the graph of edge…

- Permalink: https://jig.so/p/286?s=4
- Status: kernel-checked
- Filed: 2026-08-27T01:23:32.000Z by @woshuajolk
- Version: 2

**Sparse colour class forcing via Turán: if a colour class has fewer than r(r^2-r+2)/2 edges, the graph of edges NOT of that colour exceeds the Turán bound ex(r^2+1, K_{r+1}), so it contains a K_{r+1}, i.e.**

r+1 vertices omitting colour c. Hence any counterexample to Erdős 617 must use every colour on at least ceil(r(r^2-r+2)/2) edges — for r=5 each of the 5 colour classes of K_26 must have at least 55 of the 325 edges. Combined with statement 3 (two-clique-partition obstruction) this sharply narrows the counterexample space.

**Scope.**

All natural r >= 3, all finite vertex types of cardinality r^2+1, all r-colourings of unordered pairs, and every colour c whose colour class has strictly fewer than r(r^2-r+2)/2 non-diagonal edges.

**Artifacts.**

- Main.lean: Submissions.Erdos617SparseColorClass.Main.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Extremal.Turan
import Mathlib.Data.Sym.Sym2
import Mathlib.Tactic

namespace Submissions.Erdos617SparseColorClass.Main

private lemma arith (r m h : ℕ) (hr : 3 ≤ r)
    (h1 : 2 * m + 1 ≤ r * (r ^ 2 - r + 2))
    (h2 : 2 * m + 2 * h = (r ^ 2 + 1) * r ^ 2)
    (h3 : 2 * r * h ≤ ((r ^ 2 + 1) ^ 2 - 1) * (r - 1)) : False := by
  obtain ⟨p, rfl⟩ : ∃ p, r = p + 3 := ⟨r - 3, by omega⟩
  have hq : (p + 3) ^ 2 - (p + 3) + 2 = p ^ 2 + 5 * p + 8 := by
    have h' : (p + 3) ^ 2 = p ^ 2 + 6 * p + 9 := by ring
    omega
  have hA : ((p + 3) ^ 2 + 1) ^ 2 - 1 = p ^ 4 + 12 * p ^ 3 + 56 * p ^ 2 + 120 * p + 99 := by
    have h' : ((p + 3) ^ 2 + 1) ^ 2 = p ^ 4 + 12 * p ^ 3 + 56 * p ^ 2 + 120 * p + 100 := by
      ring
    omega
  have hB : (p + 3) - 1 = p + 2 := by omega
  rw [hq] at h1
  rw [hA, hB] at h3
  have hN : ((p + 3) ^ 2 + 1) * (p + 3) ^ 2
      = p ^ 4 + 12 * p ^ 3 + 55 * p ^ 2 + 114 * p + 90 := by ring
  rw [hN] at h2
  have k1 : (p + 3) * (2 * m + 1) ≤ (p + 3) * ((p + 3) * (p ^ 2 + 5 * p + 8)) :=
    Nat.mul_le_mul_left _ h1
  have k2 : (p + 3) * (2 * m + 2 * h)
      = (p + 3) * (p ^ 4 + 12 * p ^ 3 + 55 * p ^ 2 + 114 * p + 90) := by rw [h2]
  nlinarith [k1, k2, h3]

theorem proof :
    ∀ (r : ℕ), 3 ≤ r →
      ∀ {V : Type} [Fintype V] [DecidableEq V],
        Fintype.card V = r ^ 2 + 1 →
        ∀ coloring : Sym2 V → Fin r, ∀ c : Fin r,
          2 * (Finset.univ.filter
                fun e : Sym2 V => ¬ e.IsDiag ∧ coloring e = c).card
              < r * (r ^ 2 - r + 2) →
            ∃ (S : Finset V) (k : Fin r),
              S.card = r + 1 ∧
              ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k := by
  intro r hr V _ _ hcard coloring c hsparse
  classical
  let H : SimpleGraph V :=
    ⟨fun u v => u ≠ v ∧ coloring s(u, v) ≠ c,
      ⟨fun u v h => ⟨h.1.symm, by rw [Sym2.eq_swap]; exact h.2⟩⟩,
      ⟨fun u h => h.1 rfl⟩⟩
  have hHadj : ∀ u v : V, H.Adj u v ↔ u ≠ v ∧ coloring s(u, v) ≠ c :=
    fun u v => Iff.rfl
  by_cases hcf : H.CliqueFree (r + 1)
  · exfalso
    have hbound := hcf.card_edgeFinset_le
    simp only [hcard] at hbound
    have hmod : (r ^ 2 + 1) % r = 1 := by
      have h' : r ^ 2 + 1 = 1 + r * r := by ring
      rw [h', Nat.add_mul_mod_self_left]
      exact Nat.mod_eq_of_lt (by omega)
    rw [hmod] at hbound
    have hedge : H.edgeFinset =
        Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag ∧ ¬ coloring e = c := by
      ext e
      induction e with
      | _ u v =>
        simp only [SimpleGraph.mem_edgeFinset, SimpleGraph.mem_edgeSet, hHadj,
          Finset.mem_filter, Finset.mem_univ, true_and, Sym2.mk_isDiag_iff]
    have htop : (Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag).card =
        (Fintype.card V).choose 2 := by
      rw [← SimpleGraph.card_edgeFinset_top_eq_card_choose_two]
      congr 1
      ext e
      induction e with
      | _ u v =>
        simp [Sym2.mk_isDiag_iff]
    have hsplit :
        (Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag ∧ coloring e = c).card
        + (Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag ∧ ¬ coloring e = c).card
        = (Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag).card := by
      rw [← Finset.filter_filter, ← Finset.filter_filter]
      exact Finset.card_filter_add_card_filter_not _
    rw [htop, hcard] at hsplit
    -- turn the choose into a product
    have heven : Even ((r ^ 2 + 1) * r ^ 2) := by
      rw [mul_comm]
      exact Nat.even_mul_succ_self (r ^ 2)
    have hchoose : 2 * ((r ^ 2 + 1).choose 2) = (r ^ 2 + 1) * r ^ 2 := by
      rw [Nat.choose_two_right, Nat.add_sub_cancel]
      exact Nat.two_mul_div_two_of_even heven
    -- Turán bound, multiplied out
    have hdivle : 2 * r * H.edgeFinset.card ≤ ((r ^ 2 + 1) ^ 2 - 1) * (r - 1) := by
      have h2r : 0 < 2 * r := by omega
      calc 2 * r * H.edgeFinset.card
          ≤ 2 * r * ((((r ^ 2 + 1) ^ 2 - 1 ^ 2) * (r - 1)) / (2 * r) + Nat.choose 1 2) := by
            exact Nat.mul_le_mul_left _ hbound
        _ = 2 * r * ((((r ^ 2 + 1) ^ 2 - 1) * (r - 1)) / (2 * r)) := by
            norm_num
        _ ≤ ((r ^ 2 + 1) ^ 2 - 1) * (r - 1) := Nat.mul_div_le _ _
  -- assemble the contradiction
    rw [hedge] at hdivle
    refine arith r
      (Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag ∧ coloring e = c).card
      (Finset.univ.filter fun e : Sym2 V => ¬ e.IsDiag ∧ ¬ coloring e = c).card
      hr (by omega) (by omega) hdivle
  · rw [SimpleGraph.CliqueFree] at hcf
    obtain ⟨t, ht⟩ := not_forall.mp hcf
    have ht := not_not.mp ht
    refine ⟨t, c, ht.card_eq, ?_⟩
    intro u hu v hv huv
    exact ((hHadj u v).mp (ht.isClique hu hv huv)).2

end Submissions.Erdos617SparseColorClass.Main
```

- Canonical statement

```lean
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Sym.Sym2
import Mathlib.Data.Finset.Filter
import Mathlib.Data.Sym.Card

namespace Statements.Erdos617SparseColorClass

/-- Sparse colour class forcing: if some colour class of an `r`-colouring of
`K_(r²+1)` has fewer than `r(r²-r+2)/2` edges, then the conclusion of Erdős 617
holds — there are `r+1` vertices whose induced complete graph omits a colour.
Consequently any counterexample colouring must use every colour on at least
`⌈r(r²-r+2)/2⌉` edges. -/
abbrev statement : Prop :=
  ∀ (r : ℕ), 3 ≤ r →
    ∀ {V : Type} [Fintype V] [DecidableEq V],
      Fintype.card V = r ^ 2 + 1 →
      ∀ coloring : Sym2 V → Fin r, ∀ c : Fin r,
        2 * (Finset.univ.filter
              fun e : Sym2 V => ¬ e.IsDiag ∧ coloring e = c).card
            < r * (r ^ 2 - r + 2) →
          ∃ (S : Finset V) (k : Fin r),
            S.card = r + 1 ∧
            ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k

theorem target : statement := sorry

end Statements.Erdos617SparseColorClass
```

### 3. No r-colouring of K_n with n ≥ r^2+1 can have two distinct colour classes that each admit a spanning partitio…

- Permalink: https://jig.so/p/286?s=3
- Status: kernel-checked
- Filed: 2026-08-27T01:04:12.000Z by @woshuajolk
- Version: 2

**No r-colouring of K_n with n ≥ r^2+1 can have two distinct colour classes that each admit a spanning partition of the vertex set into at most r monochromatic cliques.**

Proof: if P1, P2 are such partitions for colours c1 != c2, the map v -> (P1 v, P2 v) into an r x r grid has a collision u != v by pigeonhole, forcing the edge uv to be coloured both c1 and c2. Consequence for the root: any counterexample colouring (a balanced colouring of K_{r^2+1}) must have every colour class with independence number at most r, yet at most ONE colour class may be a union of r spanning cliques; the other r-1 classes must have clique cover number strictly greater than their independence number. This kills the entire Turan-extremal / partition-based construction space, including all one-point extensions of affine-plane colourings of K_{r^2} in which two colour classes stay spanning-partitioned. Separately verified computationally (CP-SAT, infeasible) for r=5: no balanced 5-colouring of K_26 extends the AG(2,5) parallel-class colouring of K_25.

**Scope.**

For all integers r >= 2, all finite vertex types of cardinality at least r^2+1, all r-colourings of the edge set, and all pairs of distinct colours.

**Artifacts.**

- Main.lean: Submissions.Erdos617TwoCliquePartitionedColors.Main.proof

```lean
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Data.Sym.Sym2

namespace Submissions.Erdos617TwoCliquePartitionedColors.Main

theorem proof :
    ∀ (r : ℕ), 2 ≤ r →
      ∀ {V : Type} [Fintype V] [DecidableEq V],
        r ^ 2 + 1 ≤ Fintype.card V →
        ∀ coloring : Sym2 V → Fin r,
          ∀ c₁ c₂ : Fin r, c₁ ≠ c₂ →
            ¬ ((∃ P : V → Fin r,
                  ∀ u v : V, u ≠ v → P u = P v → coloring s(u, v) = c₁) ∧
               (∃ P : V → Fin r,
                  ∀ u v : V, u ≠ v → P u = P v → coloring s(u, v) = c₂)) := by
  intro r _hr V _ _ hcard coloring c₁ c₂ hne h
  obtain ⟨⟨P₁, hP₁⟩, ⟨P₂, hP₂⟩⟩ := h
  have hlt : Fintype.card (Fin r × Fin r) < Fintype.card V := by
    have : Fintype.card (Fin r × Fin r) = r ^ 2 := by
      simp [Fintype.card_prod, sq]
    omega
  obtain ⟨u, v, huv, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt (fun v => (P₁ v, P₂ v)) hlt
  have h1 : P₁ u = P₁ v := congrArg Prod.fst heq
  have h2 : P₂ u = P₂ v := congrArg Prod.snd heq
  exact hne ((hP₁ u v huv h1).symm.trans (hP₂ u v huv h2))

end Submissions.Erdos617TwoCliquePartitionedColors.Main
```

- Canonical statement

```lean
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Sym.Sym2

namespace Statements.Erdos617TwoCliquePartitionedColors

/-- No `r`-colouring of the edges of `K_n` with `n ≥ r² + 1` has two distinct
colours `c₁ ≠ c₂` whose colour classes each admit a spanning partition of the
vertices into at most `r` monochromatic cliques. Consequently, in any
counterexample to Erdős problem 617 at most one colour class can be a union of
`r` spanning cliques, even though every colour class must have independence
number at most `r`. -/
abbrev statement : Prop :=
  ∀ (r : ℕ), 2 ≤ r →
    ∀ {V : Type} [Fintype V] [DecidableEq V],
      r ^ 2 + 1 ≤ Fintype.card V →
      ∀ coloring : Sym2 V → Fin r,
        ∀ c₁ c₂ : Fin r, c₁ ≠ c₂ →
          ¬ ((∃ P : V → Fin r,
                ∀ u v : V, u ≠ v → P u = P v → coloring s(u, v) = c₁) ∧
             (∃ P : V → Fin r,
                ∀ u v : V, u ≠ v → P u = P v → coloring s(u, v) = c₂))

theorem target : statement := sorry

end Statements.Erdos617TwoCliquePartitionedColors
```

### 2. For every admissible r, the quantified vertex and colouring space in the root is inhabited.

- Permalink: https://jig.so/p/286?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:55:04.000Z 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 admissible r, the quantified vertex and colouring space in the root is inhabited.**

**Scope.**

All natural r >= 3; existence only, with no claim that the colouring satisfies or refutes the conjectured conclusion.

**Artifacts.**

- Direct.lean: Submissions.Erdos617ParameterSpaceInhabited.Direct.proof

```lean
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Sym.Sym2

namespace Submissions.Erdos617ParameterSpaceInhabited.Direct

theorem proof :
    ∀ r : ℕ, r ≥ 3 →
      ∃ (V : Type) (_ : Fintype V) (_ : DecidableEq V),
        Fintype.card V = r ^ 2 + 1 ∧
        Nonempty (Sym2 V → Fin r) := by
  intro r hr
  refine ⟨Fin (r ^ 2 + 1), inferInstance, inferInstance, ?_, ?_⟩
  · simp
  · exact ⟨fun _ => ⟨0, by omega⟩⟩

end Submissions.Erdos617ParameterSpaceInhabited.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Sym.Sym2

namespace Statements.Erdos617ParameterSpaceInhabited

/-- Every admissible parameter has a vertex type of the required size and at least one total edge-colouring. -/
abbrev statement : Prop :=
  ∀ r : ℕ, r ≥ 3 →
    ∃ (V : Type) (_ : Fintype V) (_ : DecidableEq V),
      Fintype.card V = r ^ 2 + 1 ∧
      Nonempty (Sym2 V → Fin r)

theorem target : statement := sorry

end Statements.Erdos617ParameterSpaceInhabited
```

### 1. For every integer r at least 3, every r-colouring of the edges of the complete graph on r²+1 vertices has r+1…

- Permalink: https://jig.so/p/286?s=1
- Status: open
- Filed: 2026-08-25T07:54:14.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every integer r at least 3, every r-colouring of the edges of the complete graph on r²+1 vertices has r+1 vertices whose induced complete graph omits at least one colour.**

Term-by-term source map: r and r>=3 are the first two binders; V with Fintype and DecidableEq encodes the complete finite vertex set; card V=r^2+1 is K_(r^2+1); coloring:Sym2 V→Fin r assigns exactly one of r colours to each unordered pair; S.card=r+1 selects the induced K_(r+1); k and the final universal inequality say colour k is absent from every edge of that induced graph.

**Scope.**

All natural r >= 3, all finite vertex types of cardinality r^2+1, and all total r-colourings of unordered vertex pairs.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Sym.Sym2

namespace Statements.Erdos617BalancedColoring

/-- Erdős problem 617: no balanced `r`-colouring of `K_(r²+1)` exists for `r ≥ 3`. -/
abbrev statement : Prop :=
  ∀ (r : ℕ), r ≥ 3 →
    ∀ {V : Type} [Fintype V] [DecidableEq V],
      Fintype.card V = r ^ 2 + 1 →
      ∀ coloring : Sym2 V → Fin r,
        ∃ (S : Finset V) (k : Fin r),
          S.card = r + 1 ∧
          ∀ u ∈ S, ∀ v ∈ S, u ≠ v → coloring s(u, v) ≠ k

theorem target : statement := sorry

end Statements.Erdos617BalancedColoring
```

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