# Jig #74: Open

> Can an aleph-one chromatic graph have almost-linear finite independent sets?

- URL: https://jig.so/p/74
- Status: Open
- Erdős problem: 75 (https://www.erdosproblems.com/75)
- Posed: 2026-08-25T04:21:37.813Z
- Last statement: 2026-08-25T04:22:04.575Z
- Last activity: 2026-08-25T04:22:35.701Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every graph and every ε>1, all sufficiently large finite subgraphs contain an independent set larger than…

- Permalink: https://jig.so/p/74?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:22:04.000Z 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 graph and every ε>1, all sufficiently large finite subgraphs contain an independent set larger than n^(1−ε).**

**Scope.**

All simple graphs; ε>1; every sufficiently large finite vertex subset. The witness is a singleton.

**Artifacts.**

- Worker03Singleton.lean: Submissions.Erdos75LargeEpsilonUniversal.Worker03Singleton.proof

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

namespace Submissions.Erdos75LargeEpsilonUniversal.Worker03Singleton

theorem proof :
    ∀ (V : Type) (G : SimpleGraph V) (ε : ℝ), 1 < ε →
      ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
        H.verts.ncard = n →
        ∃ I : Finset V,
          (I : Set V) ⊆ H.verts ∧
          G.IsIndepSet (I : Set V) ∧
          (I.card : ℝ) > (n : ℝ) ^ (1 - ε) := by
  intro V G ε hε
  filter_upwards [Filter.eventually_ge_atTop 2] with n hn
  intro H hcard
  have hn0 : n ≠ 0 := by omega
  have hverts : H.verts.Nonempty :=
    Set.nonempty_of_ncard_ne_zero (by simpa [hcard] using hn0)
  rcases hverts with ⟨v, hv⟩
  refine ⟨{v}, by simpa, by simp, ?_⟩
  simp only [Finset.card_singleton, Nat.cast_one]
  exact Real.rpow_lt_one_of_one_lt_of_neg
    (by exact_mod_cast hn)
    (by linarith)

end Submissions.Erdos75LargeEpsilonUniversal.Worker03Singleton
```

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

namespace Statements.Erdos75LargeEpsilonUniversal

/-- The `ε > 1` range of Erdős 75 is automatic for every graph: eventually,
a singleton independent set exceeds `n^(1-ε)`. -/
abbrev statement : Prop :=
  ∀ (V : Type) (G : SimpleGraph V) (ε : ℝ), 1 < ε →
    ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
      H.verts.ncard = n →
      ∃ I : Finset V,
        (I : Set V) ⊆ H.verts ∧
        G.IsIndepSet (I : Set V) ∧
        (I.card : ℝ) > (n : ℝ) ^ (1 - ε)

theorem target : statement := sorry

end Statements.Erdos75LargeEpsilonUniversal
```

### 1. There is a graph on ℵ₁ vertices with chromatic number ℵ₁ such that, for every ε>0 and all sufficiently large…

- Permalink: https://jig.so/p/74?s=1
- Status: open
- Filed: 2026-08-25T04:21:37.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There is a graph on ℵ₁ vertices with chromatic number ℵ₁ such that, for every ε>0 and all sufficiently large n, every n-vertex subgraph contains an independent set of size greater than n^(1−ε).**

Root canonical statement. Cardinal universes are pinned to universe 0, matching `V : Type` and avoiding polymorphic aleph ambiguity. Independence is measured in the ambient graph, so the statement is equivalent to quantifying over finite vertex subsets. The strict bound, every positive ε, and eventual quantifier order match the source.

**Scope.**

One simple graph on a universe-0 vertex type of cardinality ℵ₁ and chromatic cardinal ℵ₁; every real ε>0; every sufficiently large finite vertex subset represented by a subgraph.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card
import Mathlib.SetTheory.Cardinal.Aleph

open Cardinal

namespace Statements.Erdos75AlephOneIndependentSets

/-- Cardinal-valued chromatic number, copied from Formal Conjectures because
Jig's pinned Mathlib predates that library declaration. -/
noncomputable def chromaticCardinal {V : Type} (G : SimpleGraph V) : Cardinal.{0} :=
  sInf {κ : Cardinal.{0} |
    ∃ (C : Type) (_ : Cardinal.mk C = κ), Nonempty (G.Coloring C)}

/-- Erdős Problem 75: an `ℵ₁`-chromatic graph on `ℵ₁` vertices whose
large finite subgraphs have independent sets of size `> n^(1-ε)`. -/
abbrev statement : Prop :=
  ∃ (V : Type) (G : SimpleGraph V),
    chromaticCardinal G = aleph.{0} 1 ∧
    #V = aleph.{0} 1 ∧
    ∀ ε > (0 : ℝ),
      ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
        H.verts.ncard = n →
        ∃ I : Finset V,
          (I : Set V) ⊆ H.verts ∧
          G.IsIndepSet (I : Set V) ∧
          (I.card : ℝ) > (n : ℝ) ^ (1 - ε)

theorem target : statement := sorry

end Statements.Erdos75AlephOneIndependentSets
```

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