804.lean`

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 804

References:

erdosproblems.com/804
[Er91] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[AlSu07] Alon, Noga and Sudakov, Benny, On graphs with subgraphs having large independence numbers. J. Graph Theory (2007), 149-157.

Imports (1)

FormalConjecturesUtil

def Erdos804.HasLocalIndependence A graph G on n vertices in which every induced subgraph on m vertices has an independent set of size at least t line 40 source
Source documentation
A graph G on n vertices in which every induced subgraph on m vertices has an independent set of size at least t.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin
Proof uses: Finset, Eq, Finset.card, Exists, And, LE.le, SimpleGraph.IsIndepSet
6 instances, coercions and auxiliary constants left out.
```
def HasLocalIndependence {n : ℕ} (G : SimpleGraph (Fin n)) (m t : ℕ) : Prop :=
…
```
def Erdos804.f f m n is maximal such that any graph on n vertices in which every induced subgraph on m vertices has an independent set of size at least line 46 source
Source documentation
f m n is maximal such that any graph on n vertices in which every induced subgraph on m vertices has an independent set of size at least $\log n$ must contain an independent set of size at least f m n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, SimpleGraph, Fin, And, Erdos804.HasLocalIndependence, Nat.ceil, Real, Real.log, Eq, SimpleGraph.indepNum
9 instances, coercions and auxiliary constants left out.
```
noncomputable def f (m n : ℕ) : ℕ :=
…
```
theorem Erdos804.erdos_804.parts.i sorry line 61 source
depends on sorry what this means
Statement uses: Iff, False, Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HPow.hPow, HSub.hSub, HDiv.hDiv, Erdos804.f, Nat.floor, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos804.erdos_804.parts.ii sorry line 74 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HMul.hMul, HPow.hPow, Real.log, Erdos804.f, Nat.floor, Filter.atTop
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 Erdos804.erdos_804.variants.alon_sudakov sorry line 86 source
depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.log, Erdos804.f, Nat.floor, Filter.atTop
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.