Extremal.lean`

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

10 declarations, 0 contain sorry, 0 declared as axioms

Extremal parameters for hypergraphs

Shared counting and local-to-global parameters for uniform hypergraphs, designs, transversals and dense subgraphs. Finite extrema use all labeled edge families; isomorphic copies are not counted separately when counting edges or block-size profiles.

Imports (4)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Hypergraph.fixedSizeFamilies All r-uniform hypergraphs on n vertices with exactly m edges line 38 source
Source documentation
All r-uniform hypergraphs on n vertices with exactly m edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin
Proof uses: Finset.powersetCard, Finset.univ
1 instances, coercions and auxiliary constants left out.
```
def fixedSizeFamilies (n r m : ℕ) : Finset (Finset (Finset (Fin n))) :=
…
```
def Hypergraph.matchingProbability Probability of a matching in a uniformly chosen fixed-size edge family line 45 source
Source documentation
Probability of a matching in a uniformly chosen fixed-size edge family. This is the cardinality of the event divided by that of the finite sample space; it is zero when m exceeds the number of possible edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HDiv.hDiv, Finset.card, Finset, Fin, Finset.filter, Finset.HasHypergraphMatching, Classical.propDecidable, Hypergraph.fixedSizeFamilies
5 instances, coercions and auxiliary constants left out.
```
noncomputable def matchingProbability (n r m k : ℕ) : ℝ :=
…
```
def Hypergraph.linearIndependenceNumber Smallest independence number of a linear three-uniform hypergraph on n vertices line 51 source
Source documentation
Smallest independence number of a linear three-uniform hypergraph on n vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HSub.hSub, Finset.sup, Finset, Fin, Finset.filter, Finset.IsLinearHypergraph, Classical.propDecidable, Finset.powerset, Finset.powersetCard, OfNat.ofNat, Finset.univ, Finset.hypergraphIndependenceNumber
8 instances, coercions and auxiliary constants left out.
```
noncomputable def linearIndependenceNumber (n : ℕ) : ℕ :=
…
```
def Hypergraph.localTransversalBound Maximum global transversal size under a local vertex-set constraint line 57 source
Source documentation
Maximum global transversal size under a local vertex-set constraint. Infinity records that no finite bound works.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iInf, Finset, Fin, Finset.IsUniform, LE.le, Finset.card, Set.HasFiniteTransversal, Finset.hypergraphInduce
13 instances, coercions and auxiliary constants left out.
```
noncomputable def localTransversalBound (r v a : ℕ) : ℕ∞ :=
…
```
def Hypergraph.subfamilyTransversalBound Global transversal bound for a uniform family whose small subfamilies have small transversals line 65 source
Source documentation
Global transversal bound for a uniform family whose small subfamilies have small transversals. Subfamilies with fewer than q edges are included.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iInf, Set, Finset, Membership.mem, Eq, Finset.card, LE.le, Set.HasFiniteTransversal
14 instances, coercions and auxiliary constants left out.
```
noncomputable def subfamilyTransversalBound (k q a : ℕ) : ℕ∞ :=
…
```
def Hypergraph.minIntersectingEdges Least number of edges in an intersecting r-uniform family with transversal size r line 73 source
Source documentation
Least number of edges in an intersecting r-uniform family with transversal size r. Infinity records a missing witness.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iInf, Finset, Fin, Finset.IsUniform, Set.Intersecting, Set.HasFiniteTransversal, Not, HSub.hSub, OfNat.ofNat, Finset.card
18 instances, coercions and auxiliary constants left out.
```
noncomputable def minIntersectingEdges (r : ℕ) : ℕ∞ :=
…
```
def Hypergraph.sparseConfigurationThreshold Least vertex count forcing the Brown--Erdős--Sós little-o conclusion line 81 source
Source documentation
Least vertex count forcing the Brown--Erdős--Sós little-o conclusion. Infinity records the absence of a finite threshold.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, ENat
Proof uses: iInf, Asymptotics.IsLittleO, Real, Filter.atTop, Hypergraph.configurationExtremalNumber, HPow.hPow, OfNat.ofNat
17 instances, coercions and auxiliary constants left out.
```
noncomputable def sparseConfigurationThreshold (r e : ℕ) : ℕ∞ :=
…
```
def Hypergraph.edgeDensity Edge density normalized by the number of possible uniform edges line 86 source
Source documentation
Edge density normalized by the number of possible uniform edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset, Fin, Real
Proof uses: HDiv.hDiv, Finset.card, Nat.choose
5 instances, coercions and auxiliary constants left out.
```
noncomputable def edgeDensity {n : ℕ} (r : ℕ) (H : Finset (Finset (Fin n))) : ℝ :=
…
```
def Hypergraph.HasDensityJump A density increase on growing induced vertex sets for every growing host sequence line 91 source
Source documentation
A density increase on growing induced vertex sets for every growing host sequence. The boolean chooses the strict or weak lower bound at the initial density.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Bool
Proof uses: Filter.Tendsto, Filter.atTop, Finset, Fin, Finset.IsUniform, ite, Eq, Bool.true, LT.lt, Filter.liminf, Hypergraph.edgeDensity, LE.le, Exists, And, Finset.card, HDiv.hDiv, Finset.hypergraphInduce, Nat.choose
12 instances, coercions and auxiliary constants left out.
```
def HasDensityJump (r : ℕ) (α β : ℝ) (strict : Bool) : Prop :=
…
```
def Hypergraph.balancedColoringThreshold The least size above which a coloring gives both colors a positive share of every induced complete uniform hypergraph line 105 source
Source documentation
The least size above which a coloring gives both colors a positive share of every induced complete uniform hypergraph. Positivity at α = 0 retains the Ramsey convention.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, Fin, Bool, LE.le, Finset.card, Finset.filter, Eq, Finset.powersetCard, And, LT.lt, OfNat.ofNat, HMul.hMul, Nat.choose
10 instances, coercions and auxiliary constants left out.
```
noncomputable def balancedColoringThreshold (n r : ℕ) (α : ℝ) : ℕ :=
…
```