150.lean`

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

7 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 150

References:

erdosproblems.com/150
[Br24] Bradač, D., On a question of Erdős and Nešetřil about minimal cuts in a graph. arXiv:2409.02974 (2024).
[Er88] Erdős, P., Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.
[FKTV08] Fomin, Fedor V. and Kratsch, Dieter and Todinca, Ioan and Villanger, Yngve, Exact algorithms for treewidth and minimum fill-in. SIAM J. Comput. (2008), 1058-1079.
[FoVi12] Fomin, Fedor V. and Villanger, Yngve, Treewidth computation and extremal combinatorics. Combinatorica (2012), 289-308.
[GaMa18] Gaspers, Serge and Mackenzie, Simon, On the number of minimal separators in graphs. J. Graph Theory (2018), 653-659.

Imports (1)

FormalConjecturesUtil

def Erdos150.IsMinimalCut Documented declaration: IsMinimalCut line 47 source
Source documentation
A minimal cut of a graph G is a minimal set T of vertices whose removal disconnects G, that is, G - T fails to be preconnected while G - S is preconnected for every S ⊂ T.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Set
Proof uses: And, Not, SimpleGraph.Preconnected, Compl.compl, SimpleGraph.induce, LT.lt
9 instances, coercions and auxiliary constants left out.
```
def IsMinimalCut {V : Type*} (G : SimpleGraph V) (T : Set V) : Prop :=
…
```
def Erdos150.maxMinimalCuts The maximum number $c(n)$ of minimal cuts a graph on $n$ vertices can have line 51 source
Source documentation
The maximum number $c(n)$ of minimal cuts a graph on $n$ vertices can have.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, SimpleGraph, Fin, Eq, Set.ncard, Set, Erdos150.IsMinimalCut
1 instances, coercions and auxiliary constants left out.
```
noncomputable def maxMinimalCuts (n : ℕ) : ℕ :=
…
```
theorem Erdos150.erdos_150 sorry line 72 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Filter.Tendsto, HPow.hPow, Erdos150.maxMinimalCuts, HDiv.hDiv, Filter.atTop, nhds
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos150.erdos_150.variants.erdos_nesetril Asked by Erdős and Nešetřil, who also ask whether $c(3m+2)=3^m$ sorry line 84 source
Source documentation
Asked by Erdős and Nešetřil, who also ask whether $c(3m+2)=3^m$.
Note that the lower bound $1.4457\leq \alpha$ of Gaspers and Mackenzie [GaMa18] provides a negative answer to the above question of Erdős and Nešetřil.

depends on sorry what this means
Statement uses: Iff, False, Nat, Eq, Erdos150.maxMinimalCuts, HAdd.hAdd, HMul.hMul, OfNat.ofNat, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_150.variants.erdos_nesetril :
    answer(False) ↔ ∀ m : ℕ, maxMinimalCuts (3 * m + 2) = 3 ^ m := by
…
```
Mentions: Erdos150.maxMinimalCuts
theorem Erdos150.erdos_150.variants.seymour Seymour observed that $c(3m+2)\geq 3^m$, as seen by the graph of $m$ independent paths of length $4$ joining two vertices sorry line 93 source
Source documentation
Seymour observed that $c(3m+2)\geq 3^m$, as seen by the graph of $m$ independent paths of length $4$ joining two vertices.

depends on sorry what this means
Statement uses: Nat, LE.le, HPow.hPow, OfNat.ofNat, Erdos150.maxMinimalCuts, HAdd.hAdd, HMul.hMul
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_150.variants.seymour (m : ℕ) : 3 ^ m ≤ maxMinimalCuts (3 * m + 2) := by
…
```
Mentions: Erdos150.maxMinimalCuts
theorem Erdos150.erdos_150.variants.lower_bound The current best-known bounds on $\alpha$ are $$1.4457\leq \alpha \leq \frac{1+\sqrt{5}}{2}\approx 1.618.$$ The lower bound is due to Gaspers and Mackenzie [GaMa18] sorry line 102 source
Source documentation
The current best-known bounds on $\alpha$ are $$1.4457\leq \alpha \leq \frac{1+\sqrt{5}}{2}\approx 1.618.$$ The lower bound is due to Gaspers and Mackenzie [GaMa18].

depends on sorry what this means
Statement uses: Real, Filter.Tendsto, Nat, HPow.hPow, Erdos150.maxMinimalCuts, HDiv.hDiv, OfNat.ofNat, Filter.atTop, nhds, LE.le, OfScientific.ofScientific, Bool.true
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
33 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_150.variants.lower_bound (α : ℝ)
    (hα : Tendsto (fun n : ℕ ↦ (maxMinimalCuts n : ℝ) ^ (1 / n : ℝ)) atTop (𝓝 α)) :
    1.4457 ≤ α := by
…
```
Mentions: Erdos150.maxMinimalCuts
theorem Erdos150.erdos_150.variants.upper_bound Documented declaration: upper_bound sorry line 113 source
Source documentation
The current best-known bounds on $\alpha$ are $$1.4457\leq \alpha \leq \frac{1+\sqrt{5}}{2}\approx 1.618.$$ The upper bound is due to Fomin and Villanger [FoVi12] (with a simpler proof in [GaMa18]).

depends on sorry what this means
Statement uses: Real, Filter.Tendsto, Nat, HPow.hPow, Erdos150.maxMinimalCuts, HDiv.hDiv, OfNat.ofNat, Filter.atTop, nhds, LE.le, HAdd.hAdd, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_150.variants.upper_bound (α : ℝ)
    (hα : Tendsto (fun n : ℕ ↦ (maxMinimalCuts n : ℝ) ^ (1 / n : ℝ)) atTop (𝓝 α)) :
    α ≤ (1 + Real.sqrt 5) / 2 := by
…
```
Mentions: Erdos150.maxMinimalCuts