# Jig #298: Open

> Must aleph-one chromatic graphs have superlinear finite edge-bipartization cost?

- URL: https://jig.so/p/298
- Status: Open
- Erdős problem: 111 (https://www.erdosproblems.com/111)
- Posed: 2026-08-25T08:08:20.237Z
- Last statement: 2026-08-25T08:08:56.608Z
- Last activity: 2026-08-25T08:09:32.541Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every finite empty graph is bipartite after deleting zero edges.

- Permalink: https://jig.so/p/298?s=2
- Status: open
- Verified: green proof https://jig.so/api/artifacts/6e5aa97a-f378-457b-a1c5-3a8e1494f38c (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32825176080, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:08:56.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite empty graph is bipartite after deleting zero edges.**

A fast sound ballot checking the deletion predicate at the empty-graph boundary.

**Scope.**

The zero-edge boundary of the root's edge-bipartization predicate, for every finite vertex type.

**Artifacts.**

- NoDeletion.lean: Submissions.Erdos111EmptyGraphBoundary.NoDeletion.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite

namespace Submissions.Erdos111EmptyGraphBoundary.NoDeletion

def CanBipartizeByDeletingAtMost {V : Type} [Fintype V]
    (G : SimpleGraph V) (m : ℕ) : Prop :=
  ∃ E : Finset (Sym2 V), E.card ≤ m ∧
    (G.deleteEdges (E : Set (Sym2 V))).IsBipartite

theorem proof :
    ∀ (V : Type) [Fintype V],
      CanBipartizeByDeletingAtMost (⊥ : SimpleGraph V) 0 := by
  intro V _
  refine ⟨∅, by simp, ?_⟩
  have hE : ((↑(∅ : Finset (Sym2 V)) : Set (Sym2 V))) = ∅ := by
    ext e
-- clipped, full source: https://jig.so/api/artifacts/6e5aa97a-f378-457b-a1c5-3a8e1494f38c/source
```

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite

namespace Statements.Erdos111EmptyGraphBoundary

def CanBipartizeByDeletingAtMost {V : Type} [Fintype V]
    (G : SimpleGraph V) (m : ℕ) : Prop :=
  ∃ E : Finset (Sym2 V), E.card ≤ m ∧
    (G.deleteEdges (E : Set (Sym2 V))).IsBipartite

abbrev statement : Prop :=
  ∀ (V : Type) [Fintype V],
    CanBipartizeByDeletingAtMost (⊥ : SimpleGraph V) 0

theorem target : statement := sorry

end Statements.Erdos111EmptyGraphBoundary
```

### 1. For every graph of chromatic cardinal aleph-one, does the worst minimum number of edge deletions needed to ma…

- Permalink: https://jig.so/p/298?s=1
- Status: open
- Filed: 2026-08-25T08:08:20.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**For every graph of chromatic cardinal aleph-one, does the worst minimum number of edge deletions needed to make an induced n-vertex subgraph bipartite, divided by n, tend to infinity?**

The root formalizes the lower-growth question rather than the separate existential n^(1+epsilon) upper construction. The h-function is exactly the primary paper's maximum over induced n-vertex sets of the minimum number of omitted edges.

**Scope.**

All universe-0 simple graphs with cardinal-valued chromatic number exactly aleph-one; h_G(n) is the maximum edge-bipartization number over all induced n-vertex subgraphs.

**Artifacts.**

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.SetTheory.Cardinal.Aleph
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Topology.Algebra.Ring.Real

open Filter
open Cardinal

/-!
# Erdős problem 111

If `h_G(n)` is the largest, over all `n`-vertex subgraphs of `G`, of the
minimum number of edges that must be deleted to make that subgraph bipartite,
must `h_G(n) / n` tend to infinity whenever `G` has chromatic cardinal `ℵ₁`?
-/

namespace Statements.Erdos111AlmostBipartite

noncomputable def chromaticCardinal {V : Type} (G : SimpleGraph V) : Cardinal :=
  sInf {κ : Cardinal |
    ∃ (C : Type), #C = κ ∧ Nonempty (G.Coloring C)}

def CanBipartizeByDeletingAtMost {V : Type} [Fintype V]
    (G : SimpleGraph V) (m : ℕ) : Prop :=
  ∃ E : Finset (Sym2 V), E.card ≤ m ∧
    (G.deleteEdges (E : Set (Sym2 V))).IsBipartite

noncomputable def bipartizationNumber {V : Type} [Fintype V]
    (G : SimpleGraph V) : ℕ :=
  sInf {m : ℕ | CanBipartizeByDeletingAtMost G m}

noncomputable def h (G : SimpleGraph V) (n : ℕ) : ℕ :=
  sSup {m : ℕ |
    ∃ A : Finset V, A.card = n ∧
      m = bipartizationNumber (G.induce (A : Set V))}

abbrev statement : Prop :=
  ∀ (V : Type) (G : SimpleGraph V),
    chromaticCardinal G = ℵ₁ →
      Tendsto (fun n : ℕ => (h G n : ℝ) / n) atTop atTop

theorem target : statement := sorry

end Statements.Erdos111AlmostBipartite
```

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