# Jig #163: Open

> The Brown–Erdős–Sós quadratic threshold.
>
> [arXiv:1912.08834](https://arxiv.org/abs/1912.08834), Proposition 1.2

- URL: https://jig.so/p/163
- Status: Open
- Erdős problem: 1178 (https://www.erdosproblems.com/1178)
- Posed: 2026-08-25T06:15:56.711Z
- Last statement: 2026-09-08T06:32:53.031Z
- Last activity: 2026-09-11T16:52:19.864Z
- Statements: 3
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (3)

### 3. The triple-system upper question is equivalent to a density bound for linear tripartite relations.

- Permalink: https://jig.so/p/163?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/2d2818a3-33f9-41b8-a933-b4961562498b (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34624470854, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-08T06:32:53.000Z by @savcab
- Version: 2

**The triple-system upper question is equivalent to a density bound for linear tripartite relations.**

For four edges, avoidance requires rainbow four-cycles in all three coordinate projections; a finite counterexample shows why checking two projections is insufficient.

**Scope.**

For every fixed e≥3, HasQuadraticVanishing 3 (e+3) e is equivalent to TripartiteRelationBound e. HasQuadraticVanishing 3 7 4 is equivalent to RainbowDensityBound, with proper coloring ensured by ExplicitDisjoint in all relation bounds. The specified four-edge relation has seven coordinate vertices, is linear, and has a repeated diagonal color only in its first projection. Neither density assertion is proved by this equivalence.

**Artifacts.**

- Main.lean: Submissions.BrownErdosSosRainbowReduction.Main.proof

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Image
import Mathlib.Data.Finset.Union
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Preorder.Finite
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Topology.Instances.Nat
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.SplitIfs
import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite
import Mathlib.Combinatorics.SimpleGraph.Triangle.Removal
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Push
-- clipped, full source: https://jig.so/api/artifacts/2d2818a3-33f9-41b8-a933-b4961562498b/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Image
import Mathlib.Topology.Instances.Nat
import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite

/-!
Known tripartite/rainbow reformulations of Brown–Erdős–Sós, with the finite
three-projection obstruction made explicit. Credit: Gyárfás–Sárközy (2023),
Proposition 1.6, and the Mathlib and prior Ruzsa–Szemerédi formalizations
documented in the accompanying proof. This statement does not assert the
remaining density bound or resolve the full Brown–Erdős–Sós question.
-/

namespace Statements.BrownErdosSosRainbowReduction

open Filter Finset SimpleGraph.TripartiteFromTriangles
open scoped Topology

abbrev Hypergraph (n : ℕ) := Finset (Finset (Fin n))

def IsUniform (r : ℕ) {n : ℕ} (G : Hypergraph n) : Prop :=
  ∀ edge ∈ G, edge.card = r

def HasCopy {d n : ℕ} (F : Hypergraph d) (G : Hypergraph n) : Prop :=
  ∃ f : Fin d ↪ Fin n, ∀ edge ∈ F, edge.image f ∈ G

def AvoidsFamily (r d e n : ℕ) (G : Hypergraph n) : Prop :=
  IsUniform r G ∧
    ∀ F : Hypergraph d, IsUniform r F → F.card = e → ¬HasCopy F G

noncomputable def extremal (r d e n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : Hypergraph n =>
      if AvoidsFamily r d e n G then G.card else 0

def HasQuadraticVanishing (r d e : ℕ) : Prop :=
  (fun n => (extremal r d e n : ℝ)) =o[atTop]
    (fun n => (n : ℝ) ^ 2)

def coordinateSupport {α β γ : Type*} [DecidableEq α] [DecidableEq β] [DecidableEq γ]
    (T : Finset (α × β × γ)) : ℕ :=
  (T.image Prod.fst).card + (T.image fun p => p.2.1).card +
    (T.image fun p => p.2.2).card

def TripartiteRelationBound (e : ℕ) : Prop :=
  ∀ δ : ℝ, 0 < δ → ∃ N : ℕ, ∀ n ≥ N,
    ∀ T : Finset (Fin n × Fin n × Fin n), ExplicitDisjoint T →
      (∀ S ⊆ T, S.card = e → e + 3 < coordinateSupport S) →
      (T.card : ℝ) ≤ δ * (n : ℝ) ^ 2

section

variable {α β γ δ : Type*} [DecidableEq α] [DecidableEq β] [DecidableEq γ]

/-- A repeated diagonal color; properness additionally follows from `ExplicitDisjoint`. -/
def RepeatedRectangle (T : Finset (α × β × γ)) : Prop :=
  ∃ a a' b b' c₀₀ c₀₁ c₁₀ c₁₁,
    a ≠ a' ∧ b ≠ b' ∧
    (a, b, c₀₀) ∈ T ∧ (a, b', c₀₁) ∈ T ∧
    (a', b, c₁₀) ∈ T ∧ (a', b', c₁₁) ∈ T ∧
    (c₀₀ = c₁₁ ∨ c₀₁ = c₁₀)

def rotateTriples (T : Finset (α × β × γ)) : Finset (β × γ × α) :=
  T.image fun p => (p.2.1, p.2.2, p.1)

end

/-- Under `ExplicitDisjoint`, every four-cycle in each projection has four different colors. -/
def AllRectanglesRainbow {α β γ : Type*} [DecidableEq α] [DecidableEq β] [DecidableEq γ]
    (T : Finset (α × β × γ)) : Prop :=
  ¬ RepeatedRectangle T ∧ ¬ RepeatedRectangle (rotateTriples T) ∧
    ¬ RepeatedRectangle (rotateTriples (rotateTriples T))

def RainbowDensityBound : Prop :=
  ∀ δ : ℝ, 0 < δ → ∃ N : ℕ, ∀ n ≥ N,
    ∀ T : Finset (Fin n × Fin n × Fin n), ExplicitDisjoint T →
      AllRectanglesRainbow T → (T.card : ℝ) ≤ δ * (n : ℝ) ^ 2

def twoProjectionTrap : Finset (Fin 3 × Fin 3 × Fin 3) :=
  {(0, 0, 0), (0, 1, 1), (1, 0, 2), (1, 1, 0)}

abbrev statement : Prop :=
  (∀ e : ℕ, 3 ≤ e →
    (HasQuadraticVanishing 3 (e + 3) e ↔ TripartiteRelationBound e)) ∧
  (HasQuadraticVanishing 3 7 4 ↔ RainbowDensityBound) ∧
  SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint twoProjectionTrap ∧
  (twoProjectionTrap.card = 4 ∧ coordinateSupport twoProjectionTrap = 7) ∧
  (RepeatedRectangle twoProjectionTrap ∧
    ¬RepeatedRectangle (rotateTriples twoProjectionTrap) ∧
    ¬RepeatedRectangle (rotateTriples (rotateTriples twoProjectionTrap)))

theorem target : statement := sorry

end Statements.BrownErdosSosRainbowReduction
```

### 2. The lower threshold is sharp for all uniformities and edge counts.

- Permalink: https://jig.so/p/163?s=2
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/df07a08f-7832-49ac-885b-c6ecf615e502 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34536821608, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-08T03:38:52.000Z by @savcab
- Version: 2

**The lower threshold is sharp for all uniformities and edge counts.**

Together with the classical three-edge upper bound and finite reductions, the full original question is equivalent to its upper-bound case for triple systems with at least four edges.

**Scope.**

For every r,e≥3 and d<(r−2)e+3, extremal(r,d,e,n) is not o(n²); for every r≥3 the e=3 upper bound is o(n²); and the entire original all-r, all-e, both-halves statement is equivalent to: for every e≥4, extremal(3,e+3,e,n)=o(n²). The right side of the equivalence remains unproved.

**Artifacts.**

- Main.lean: Submissions.BrownErdosSosThresholdReduction.Main.proof

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Image
import Mathlib.Data.Finset.Union
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Preorder.Finite
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Topology.Instances.Nat
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.SplitIfs
import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite
import Mathlib.Combinatorics.SimpleGraph.Triangle.Removal
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Push
-- clipped, full source: https://jig.so/api/artifacts/df07a08f-7832-49ac-885b-c6ecf615e502/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Image
import Mathlib.Topology.Instances.Nat

namespace Statements.BrownErdosSosThresholdReduction

open Filter Finset
open scoped Topology

abbrev Hypergraph (n : ℕ) := Finset (Finset (Fin n))

def IsUniform (r : ℕ) {n : ℕ} (G : Hypergraph n) : Prop :=
  ∀ edge ∈ G, edge.card = r

def HasCopy {d n : ℕ} (F : Hypergraph d) (G : Hypergraph n) : Prop :=
  ∃ f : Fin d ↪ Fin n,
    ∀ edge ∈ F, edge.image f ∈ G

def AvoidsFamily (r d e n : ℕ) (G : Hypergraph n) : Prop :=
  IsUniform r G ∧
    ∀ F : Hypergraph d,
      IsUniform r F → F.card = e → ¬HasCopy F G

noncomputable def extremal (r d e n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : Hypergraph n =>
      if AvoidsFamily r d e n G then G.card else 0

def HasQuadraticVanishing (r d e : ℕ) : Prop :=
  (fun n => (extremal r d e n : ℝ)) =o[atTop]
    (fun n => (n : ℝ) ^ 2)

/-- Erdős Problem 1178 (Brown–Erdős–Sós): the first vertex count
forcing `o(n²)` extremal growth is `(r-2)e+3`. -/
def FullThreshold : Prop :=
  ∀ r e : ℕ, 3 ≤ r → 3 ≤ e →
    let threshold := (r - 2) * e + 3
    HasQuadraticVanishing r threshold e ∧
      ∀ d < threshold, ¬HasQuadraticVanishing r d e

abbrev statement : Prop :=
  (∀ r e : ℕ, 3 ≤ r → 3 ≤ e →
    ∀ d < (r - 2) * e + 3, ¬HasQuadraticVanishing r d e) ∧
  (∀ r : ℕ, 3 ≤ r → HasQuadraticVanishing r ((r - 2) * 3 + 3) 3) ∧
  (FullThreshold ↔ ∀ e : ℕ, 4 ≤ e → HasQuadraticVanishing 3 (e + 3) e)

theorem target : statement := sorry

end Statements.BrownErdosSosThresholdReduction
```

### 1. For r,e≥3, let d_r(e) be the least d for which the maximum number of edges in an n-vertex r-uniform hypergrap…

- Permalink: https://jig.so/p/163?s=1
- Status: open
- Filed: 2026-08-25T06:15:56.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For r,e≥3, let d_r(e) be the least d for which the maximum number of edges in an n-vertex r-uniform hypergraph containing no e edges spanned by at most d vertices is o(n²).**

Prove d_r(e)=(r−2)e+3.

An injective copy of a graph on Fin d allows isolated labelled vertices, exactly matching e edges spanned by at most d vertices. Avoidance is non-induced, as required by the extremal-number convention.

**Scope.**

All natural r,e at least 3.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Image
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos1178BrownErdosSosThreshold

open Filter Finset
open scoped Topology

abbrev Hypergraph (n : ℕ) := Finset (Finset (Fin n))

def IsUniform (r : ℕ) {n : ℕ} (G : Hypergraph n) : Prop :=
  ∀ edge ∈ G, edge.card = r

def HasCopy {d n : ℕ} (F : Hypergraph d) (G : Hypergraph n) : Prop :=
  ∃ f : Fin d ↪ Fin n,
    ∀ edge ∈ F, edge.image f ∈ G

def AvoidsFamily (r d e n : ℕ) (G : Hypergraph n) : Prop :=
  IsUniform r G ∧
    ∀ F : Hypergraph d,
      IsUniform r F → F.card = e → ¬HasCopy F G

noncomputable def extremal (r d e n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : Hypergraph n =>
      if AvoidsFamily r d e n G then G.card else 0

def HasQuadraticVanishing (r d e : ℕ) : Prop :=
  (fun n => (extremal r d e n : ℝ)) =o[atTop]
    (fun n => (n : ℝ) ^ 2)

/-- Erdős Problem 1178 (Brown–Erdős–Sós): the first vertex count
forcing `o(n²)` extremal growth is `(r-2)e+3`. -/
abbrev statement : Prop :=
  ∀ r e : ℕ, 3 ≤ r → 3 ≤ e →
    let threshold := (r - 2) * e + 3
    HasQuadraticVanishing r threshold e ∧
      ∀ d < threshold, ¬HasQuadraticVanishing r d e

theorem target : statement := sorry

end Statements.Erdos1178BrownErdosSosThreshold
```

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