619.lean`

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

8 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 619

References:

erdosproblems.com/619
[EGR98] Erdős, Paul and Gyárfás, András and Ruszinkó, Miklós, How to decrease the diameter of triangle-free graphs. Combinatorica 18 (1998), 493-501.
[Er99] Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. 8 (1999), 1-6.

Imports (1)

FormalConjecturesUtil

def Erdos619.minNewEdges For a graph line 50 source
Source documentation
For a graph $G$, minNewEdges r G is the smallest number of edges that need to be added to $G$ so that it has diameter at most $r$, while preserving the property of being triangle-free. Erdős, Gyárfás and Ruszinkó denote this quantity by $h_r(G)$ in [EGR98].
Adding edges is formalised as passing to a supergraph H ≥ G on the same vertex set, the number of added edges being the (finite) cardinality of the edge set of H \ G. We use the extended diameter SimpleGraph.ediam (valued in ℕ∞) so that disconnected supergraphs, which have infinite diameter, do not qualify. If no triangle-free supergraph of G of diameter at most r exists, then minNewEdges r G = 0 by the Nat.sInf convention on the empty set; erdos_619.variants.add_edges_diam_three shows this degenerate case does not occur for connected triangle-free G when 3 ≤ r.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph
Proof uses: InfSet.sInf, Set.ofPred, Exists, And, LE.le, SimpleGraph.CliqueFree, OfNat.ofNat, ENat, SimpleGraph.ediam, Eq, Set.ncard, Sym2, SimpleGraph.edgeSet, SDiff.sdiff
7 instances, coercions and auxiliary constants left out.
```
noncomputable def minNewEdges {V : Type*} (r : ℕ) (G : SimpleGraph V) : ℕ :=
…
```
theorem Erdos619.erdos_619 sorry line 70 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Fintype, SimpleGraph, SimpleGraph.Connected, SimpleGraph.CliqueFree, Nat, LT.lt, Erdos619.minNewEdges, HMul.hMul, HSub.hSub, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos619.erdos_619.variants.add_edges_diam_three Erdős, Gyárfás and Ruszinkó [EGR98]: every connected triangle-free graph on a finite vertex set can be extended, by adding edges, to a triangle-free graph of diameter at most $3$ sorry line 87 source
Source documentation
Erdős, Gyárfás and Ruszinkó [EGR98]: every connected triangle-free graph on a finite vertex set can be extended, by adding edges, to a triangle-free graph of diameter at most $3$. This shows that the infimum defining minNewEdges r G ranges over a nonempty set for every 3 ≤ r.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, SimpleGraph.Connected, SimpleGraph.CliqueFree, OfNat.ofNat, Nat, Exists, And, LE.le, ENat, SimpleGraph.ediam
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_619.variants.add_edges_diam_three {V : Type*} [Fintype V]
    (G : SimpleGraph V) (hG : G.Connected) (hG' : G.CliqueFree 3) :
    ∃ H : SimpleGraph V, G ≤ H ∧ H.CliqueFree 3 ∧ H.ediam ≤ 3 := by
…
```
theorem Erdos619.erdos_619.variants.h_three_le Erdős, Gyárfás and Ruszinkó [EGR98]: $h_3(G) \leq n$ for every connected triangle-free graph $G$ on $n$ vertices sorry line 97 source
Source documentation
Erdős, Gyárfás and Ruszinkó [EGR98]: $h_3(G) \leq n$ for every connected triangle-free graph $G$ on $n$ vertices.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, SimpleGraph.Connected, SimpleGraph.CliqueFree, OfNat.ofNat, Nat, LE.le, Erdos619.minNewEdges, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_619.variants.h_three_le {V : Type*} [Fintype V]
    (G : SimpleGraph V) (hG : G.Connected) (hG' : G.CliqueFree 3) :
    minNewEdges 3 G ≤ Fintype.card V := by
…
```
Mentions: Erdos619.minNewEdges
theorem Erdos619.erdos_619.variants.h_five_le Erdős, Gyárfás and Ruszinkó [EGR98]: $h_5(G) \leq \frac{n-1}{2}$ for every connected triangle-free graph $G$ on $n$ vertices sorry line 107 source
Source documentation
Erdős, Gyárfás and Ruszinkó [EGR98]: $h_5(G) \leq \frac{n-1}{2}$ for every connected triangle-free graph $G$ on $n$ vertices.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, SimpleGraph.Connected, SimpleGraph.CliqueFree, OfNat.ofNat, Nat, LE.le, Real, Erdos619.minNewEdges, HDiv.hDiv, HSub.hSub, Fintype.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_619.variants.h_five_le {V : Type*} [Fintype V]
    (G : SimpleGraph V) (hG : G.Connected) (hG' : G.CliqueFree 3) :
    (minNewEdges 5 G : ℝ) ≤ (Fintype.card V - 1) / 2 := by
…
```
Mentions: Erdos619.minNewEdges
theorem Erdos619.erdos_619.variants.h_three_lower Erdős, Gyárfás and Ruszinkó [EGR98]: there is a constant $c > 0$ such that for every $n$ there exist connected triangle-free graphs $G$ on $n$ vertices with $h_3(G) \geq n - c$ sorry line 118 source
Source documentation
Erdős, Gyárfás and Ruszinkó [EGR98]: there is a constant $c > 0$ such that for every $n$ there exist connected triangle-free graphs $G$ on $n$ vertices with $h_3(G) \geq n - c$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LT.lt, Fintype, SimpleGraph, Eq, Fintype.card, SimpleGraph.Connected, SimpleGraph.CliqueFree, LE.le, HSub.hSub, Erdos619.minNewEdges
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_619.variants.h_three_lower :
    ∃ c > (0 : ℝ), ∀ n : ℕ, 0 < n →
      ∃ (V : Type) (_ : Fintype V) (G : SimpleGraph V),
        Fintype.card V = n ∧ G.Connected ∧ G.CliqueFree 3 ∧
        (n : ℝ) - c ≤ minNewEdges 3 G := by
…
```
Mentions: c, Erdos619.minNewEdges
theorem Erdos619.erdos_619.test.minNewEdges_eq_zero A triangle-free graph that already has diameter at most r needs no new edges line 131 source
Source documentation
A triangle-free graph that already has diameter at most r needs no new edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, SimpleGraph.CliqueFree, OfNat.ofNat, LE.le, ENat, SimpleGraph.ediam, Eq, Erdos619.minNewEdges
Proof uses: Iff.mpr, InfSet.sInf, Set.ofPred, Exists, And, Set.ncard, Sym2, SimpleGraph.edgeSet, SDiff.sdiff, Or, Membership.mem, Set, EmptyCollection.emptyCollection, Nat.sInf_eq_zero, Or.inl, Exists.intro, And.intro, le_rfl, of_eq_true, Eq.trans and 8 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_619.test.minNewEdges_eq_zero {V : Type*} {r : ℕ} {G : SimpleGraph V}
    (hG : G.CliqueFree 3) (hdiam : G.ediam ≤ r) :
    minNewEdges r G = 0 :=
…
```
Mentions: Erdos619.minNewEdges
theorem Erdos619.erdos_619.test.minNewEdges_singleton The graph with a single vertex is triangle-free and has diameter 0, so it needs no new edges to reach diameter at most 4 line 139 source
Source documentation
The graph with a single vertex is triangle-free and has diameter 0, so it needs no new edges to reach diameter at most 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos619.minNewEdges, Fin, OfNat.ofNat, Bot.bot, SimpleGraph
Proof uses: Erdos619.erdos_619.test.minNewEdges_eq_zero, SimpleGraph.cliqueFree_bot, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, of_eq_true, LE.le, ENat, SimpleGraph.ediam, Eq.trans, True, congrFun', congrArg, SimpleGraph.ediam_eq_zero_of_subsingleton
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_619.test.minNewEdges_singleton :
    minNewEdges 4 (⊥ : SimpleGraph (Fin 1)) = 0 :=
…
```
Mentions: Erdos619.minNewEdges