184.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 184

References:

erdosproblems.com/184
[BM22] Bucić, M. and Montgomery, R., Towards the Erdős-Gallai Cycle Decomposition Conjecture. arXiv:2211.07689 (2022).
[CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures Algorithms (2014), 608-626.
[EGP66] Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112.
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[Py85] Pyber, L., An Erdős-Gallai conjecture. Combinatorica (1985), 67-79.

Imports (1)

FormalConjecturesUtil

def Erdos184.IsCycleOrEdge A graph $H$ is a cycle or an edge if it is connected and 2-regular, or if it has exactly one edge line 45 source
Source documentation
A graph $H$ is a cycle or an edge if it is connected and 2-regular, or if it has exactly one edge.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph
Proof uses: Or, And, SimpleGraph.Connected, SimpleGraph.IsRegularOfDegree, Membership.mem, Set, SimpleGraph.neighborSet, Classical.propDecidable, SimpleGraph.Adj, OfNat.ofNat, Nat, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset
6 instances, coercions and auxiliary constants left out.
```
def IsCycleOrEdge {U : Type*} [Fintype U] (H : SimpleGraph U) : Prop :=
…
```
theorem Erdos184.erdos_184 Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges sorry line 54 source
Source documentation
Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Asymptotics.IsBigO, Filter.atTop, Fintype, DecidableEq, SimpleGraph, Finset, SimpleGraph.Subgraph, Membership.mem, Erdos184.IsCycleOrEdge, SimpleGraph.Subgraph.verts, Set, Classical.propDecidable, SimpleGraph.Subgraph.coe, SimpleGraph.IsDecomposition, LE.le, Finset.card and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_184 :
    ∃ f : ℕ → ℝ,
      (f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧
      ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      ∃ (D : Finset G.Subgraph),
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V) := by
…
```
Mentions: Erdos184.IsCycleOrEdge
theorem Erdos184.erdos_184.variants.n_log_n Erdős and Gallai [EGP66] proved that $O(n \log n)$ many cycles and edges suffices sorry line 69 source
Source documentation
Erdős and Gallai [EGP66] proved that $O(n \log n)$ many cycles and edges suffices.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Asymptotics.IsBigO, Filter.atTop, HMul.hMul, Real.log, Fintype, DecidableEq, SimpleGraph, Finset, SimpleGraph.Subgraph, Membership.mem, Erdos184.IsCycleOrEdge, SimpleGraph.Subgraph.verts, Set, Classical.propDecidable, SimpleGraph.Subgraph.coe, SimpleGraph.IsDecomposition and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_184.variants.n_log_n :
    ∃ f : ℕ → ℝ,
      (f =O[atTop] fun n : ℕ ↦ (n : ℝ) * Real.log (n : ℝ)) ∧
      ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      ∃ (D : Finset G.Subgraph),
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V) := by
…
```
Mentions: Erdos184.IsCycleOrEdge
theorem Erdos184.erdos_184.variants.lower_bound The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$ sorry line 85 source
Source documentation
The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, SimpleGraph.fromRel, LT.lt, LE.le, Finset, SimpleGraph.Subgraph, Membership.mem, Erdos184.IsCycleOrEdge, SimpleGraph.Subgraph.verts, Set, Classical.propDecidable, SimpleGraph.Subgraph.coe and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_184.variants.lower_bound :
    ∃ c > 0, ∀ᶠ n in atTop,
      let G : SimpleGraph (Fin n) := fromRel (fun (i j : Fin n) => (i : ℕ) < 3 ∧ 3 ≤ (j : ℕ));
…
```
Mentions: c
theorem Erdos184.erdos_184.variants.covering In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint sorry line 100 source
Source documentation
In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint. Pyber [Py85] proved this.

depends on sorry what this means
Statement uses: Iff, True, Fintype, DecidableEq, Nonempty, SimpleGraph, Exists, Finset, SimpleGraph.Subgraph, And, Membership.mem, Erdos184.IsCycleOrEdge, SimpleGraph.Subgraph.verts, Set, Classical.propDecidable, SimpleGraph.Subgraph.coe, Eq, Sym2, Set.iUnion, SimpleGraph.Subgraph.edgeSet and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_184.variants.covering :
    answer(True) ↔
      ∀ {V : Type} [Fintype V] [DecidableEq V] [Nonempty V] (G : SimpleGraph V),
      ∃ (D : Finset G.Subgraph),
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        (⋃ H ∈ D, H.edgeSet) = G.edgeSet ∧
        (D.card : ℝ) ≤ (Fintype.card V : ℝ) - 1 := by
…
```
Mentions: Erdos184.IsCycleOrEdge
theorem Erdos184.erdos_184.variants.bucic_montgomery The best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\log^* n)$ many cycles and edges suffice, where $\log^*$ is the iterated logarithm function sorry line 115 source
Source documentation
The best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\log^* n)$ many cycles and edges suffice, where $\log^*$ is the iterated logarithm function.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Asymptotics.IsBigO, Filter.atTop, HMul.hMul, Real.iteratedLog, Fintype, DecidableEq, SimpleGraph, Finset, SimpleGraph.Subgraph, Membership.mem, Erdos184.IsCycleOrEdge, SimpleGraph.Subgraph.verts, Set, Classical.propDecidable, SimpleGraph.Subgraph.coe, SimpleGraph.IsDecomposition and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_184.variants.bucic_montgomery :
    ∃ f : ℕ → ℝ,
      (f =O[atTop] fun n : ℕ ↦ (n : ℝ) * (Real.iteratedLog (n : ℝ) : ℝ)) ∧
      ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      ∃ (D : Finset G.Subgraph),
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V) := by
…
```
Mentions: Real.iteratedLog, Erdos184.IsCycleOrEdge
theorem Erdos184.erdos_184.variants.conlon_fox_sudakov Conlon, Fox, and Sudakov [CFS14] proved that $O_\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\epsilon n$, for any $\epsilon>0$ sorry line 131 source
Source documentation
Conlon, Fox, and Sudakov [CFS14] proved that $O_\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\epsilon n$, for any $\epsilon>0$.

depends on sorry what this means
Statement uses: Real, GT.gt, OfNat.ofNat, Exists, Nat, And, Asymptotics.IsBigO, Filter.atTop, Fintype, DecidableEq, SimpleGraph, GE.ge, SimpleGraph.minDegree, Classical.propDecidable, SimpleGraph.Adj, HMul.hMul, Fintype.card, Finset, SimpleGraph.Subgraph, Membership.mem and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_184.variants.conlon_fox_sudakov :
    ∀ ε > 0, ∃ f : ℕ → ℝ,
      (f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧
      ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      (G.minDegree : ℝ) ≥ ε * (Fintype.card V : ℝ) →
      ∃ (D : Finset G.Subgraph),
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V) := by
…
```
Mentions: Erdos184.IsCycleOrEdge