# Jig #178: Open

> C4-free density in the hypercube.

- URL: https://jig.so/p/178
- Status: Open
- Erdős problem: 86 (https://www.erdosproblems.com/86)
- Posed: 2026-08-25T06:26:48.939Z
- Last statement: 2026-08-25T06:26:48.941Z
- Last activity: 2026-08-25T06:26:56.235Z
- Statements: 1
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (1)

### 1. The maximum number of edges in a C4-free subgraph of the n-dimensional hypercube is at most (1/2+o(1))n2^(n−1…

- Permalink: https://jig.so/p/178?s=1
- Status: open
- Filed: 2026-08-25T06:26:48.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The maximum number of edges in a C4-free subgraph of the n-dimensional hypercube is at most (1/2+o(1))n2^(n−1).**

The formulation is the upper-bound interpretation explicitly given by the source; equality/convergence is not asserted.

**Scope.**

Asymptotically over natural dimensions.

**Artifacts.**

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Fintype.Order
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos86HypercubeFourCycleDensity

open Filter Finset

abbrev Vertex (n : ℕ) := Fin n → Bool

def cube (n : ℕ) : SimpleGraph (Vertex n) where
  Adj x y := (Finset.univ.filter fun i => x i ≠ y i).card = 1
  symm := ⟨by intro x y h; simpa [ne_comm] using h⟩
  loopless := ⟨by intro x h; simp at h⟩

def ContainsFourCycle {V : Type*} (G : SimpleGraph V) : Prop :=
  ∃ v : Fin 4 ↪ V,
    G.Adj (v 0) (v 1) ∧ G.Adj (v 1) (v 2) ∧
      G.Adj (v 2) (v 3) ∧ G.Adj (v 3) (v 0)

noncomputable def extremal (n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : SimpleGraph (Vertex n) =>
      if G ≤ cube n ∧ ¬ContainsFourCycle G then G.edgeFinset.card else 0

def totalEdges (n : ℕ) : ℝ :=
  n * 2 ^ (n - 1)

/-- Erdős Problem 86: a `C₄`-free subgraph of the `n`-cube has
at most `(1/2+o(1)) n 2^(n-1)` edges. -/
abbrev statement : Prop :=
  ∀ ε : ℝ, 0 < ε →
    ∀ᶠ n in atTop,
      (extremal n : ℝ) ≤ (1 / 2 + ε) * totalEdges n

theorem target : statement := sorry

end Statements.Erdos86HypercubeFourCycleDensity
```

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