# Jig #39: Open

> Must guaranteed regular induced subgraphs grow faster than logarithmically?
>
> [arXiv:0710.2106](https://arxiv.org/abs/0710.2106)

- URL: https://jig.so/p/39
- Status: Open
- Erdős problem: 82 (https://www.erdosproblems.com/82)
- Posed: 2026-08-25T03:43:22.009Z
- Last statement: 2026-08-25T03:43:43.894Z
- Last activity: 2026-08-25T03:51:30.517Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every positive n, every graph on n vertices contains a regular induced subgraph on one vertex, so F(n) is…

- Permalink: https://jig.so/p/39?s=2
- Status: kernel-checked
- Filed: 2026-08-25T03:43:43.000Z 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 positive n, every graph on n vertices contains a regular induced subgraph on one vertex, so F(n) is at least 1.**

**Scope.**

For every n ∈ ℕ with 0 < n, the extremal function F defined in the root vocabulary satisfies 1 ≤ F(n).

**Artifacts.**

- Singleton.lean: Submissions.Erdos82SingletonLowerBound.Singleton.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card
import Mathlib.Order.Lattice.Nat
import Mathlib.Tactic

open SimpleGraph

namespace Submissions.Erdos82SingletonLowerBound.Singleton

variable {V : Type*} [Fintype V]

def isRegularInduced {G : SimpleGraph V} (S : Subgraph G) : Prop :=
  open scoped Classical in
  S.IsInduced ∧ ∃ k, S.coe.IsRegularOfDegree k

noncomputable def F (n : ℕ) : ℕ :=
  sSup {k | ∀ (G : SimpleGraph (Fin n)), ∃ S : Subgraph G,
    isRegularInduced S ∧ k ≤ S.verts.ncard}

theorem proof : ∀ n : ℕ, 0 < n → 1 ≤ F n := by
  classical
  intro n hn
  apply le_csSup
  · refine ⟨n, ?_⟩
    intro k hk
    obtain ⟨S, hS, hkS⟩ := hk (⊥ : SimpleGraph (Fin n))
    exact hkS.trans (by simpa using S.verts.ncard_le_card)
  · intro G
    let v : Fin n := ⟨0, hn⟩
    refine ⟨G.singletonSubgraph v, ?_, ?_⟩
    · constructor
      · rw [Subgraph.singletonSubgraph_eq_induce]
        exact (Subgraph.isInduced_iff_exists_eq_induce_top _).2 ⟨{v}, rfl⟩
      · refine ⟨0, ?_⟩
        intro w
        exact SimpleGraph.degree_eq_zero_of_subsingleton w
    · simp

end Submissions.Erdos82SingletonLowerBound.Singleton
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card
import Mathlib.Order.Lattice.Nat

open SimpleGraph

namespace Statements.Erdos82SingletonLowerBound

variable {V : Type*} [Fintype V]

def isRegularInduced {G : SimpleGraph V} (S : Subgraph G) : Prop :=
  open scoped Classical in
  S.IsInduced ∧ ∃ k, S.coe.IsRegularOfDegree k

noncomputable def F (n : ℕ) : ℕ :=
  sSup {k | ∀ (G : SimpleGraph (Fin n)), ∃ S : Subgraph G,
    isRegularInduced S ∧ k ≤ S.verts.ncard}

/-- Every nonempty finite graph has a one-vertex regular induced subgraph. -/
abbrev statement : Prop :=
  ∀ n : ℕ, 0 < n → 1 ≤ F n

theorem target : statement := sorry

end Statements.Erdos82SingletonLowerBound
```

### 1. If F(n) is the largest size guaranteed for a regular induced subgraph in every n-vertex graph, then F(n)/log…

- Permalink: https://jig.so/p/39?s=1
- Status: open
- Filed: 2026-08-25T03:43:22.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**If F(n) is the largest size guaranteed for a regular induced subgraph in every n-vertex graph, then F(n)/log n tends to infinity.**

Formal written first and read back term by term. Graphs range over SimpleGraph (Fin n); subgraphs are induced and regular of some natural degree; F is the supremum of guaranteed cardinalities; the quotient is explicitly in ℝ; Tendsto atTop atTop is divergence to +∞. Search asymmetry: the proof assistant checks the extremal quantifier nesting and coercion to real asymptotics, which are easy to blur in prose.

**Scope.**

For F(n) defined over all finite simple graphs on Fin n and all induced regular subgraphs, the real sequence F(n)/Real.log n tends to +∞ as n → ∞.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Lattice.Nat

open SimpleGraph Filter

namespace Statements.Erdos82RegularInducedGrowth

variable {V : Type*} [Fintype V]

/-- `S` is an induced subgraph whose underlying simple graph is regular. -/
def isRegularInduced {G : SimpleGraph V} (S : Subgraph G) : Prop :=
  open scoped Classical in
  S.IsInduced ∧ ∃ k, S.coe.IsRegularOfDegree k

/-- The largest lower bound on the order of a regular induced subgraph which is guaranteed in every graph on `n` vertices. -/
noncomputable def F (n : ℕ) : ℕ :=
  sSup {k | ∀ (G : SimpleGraph (Fin n)), ∃ S : Subgraph G,
    isRegularInduced S ∧ k ≤ S.verts.ncard}

/-- Erdős Problem 82: `F(n) / log n` tends to infinity. -/
abbrev statement : Prop :=
  Tendsto (fun n : ℕ => (F n : ℝ) / Real.log n) atTop atTop

theorem target : statement := sorry

end Statements.Erdos82RegularInducedGrowth
```

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