Ramsey.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

Ramsey-type properties of graph pairs

The Erdős–Hajnal "exceptional pair" trio of predicates:

HasFiniteRamseyProperty G₁ G₂ — for every n ≥ 1 there is a G₁-free graph H such that every n-edge-colouring of H contains a monochromatic G₂.
HasCountableRamseyEscape G₁ G₂ — every G₁-free graph H admits a countable edge-colouring with every colour class G₂-free.
IsErdosHajnalExceptional G₁ G₂ — the conjunction of both.

These were introduced for Erdős Problem 596 but are reusable for Problem 595 and other Ramsey-type questions; we factor them out per mo271's review.

It also defines the two-colour graph Ramsey number graphRamsey G H and the notion of a Ramsey size linear graph IsRamseySizeLinear G of Erdős, Faudree, Rousseau and Schelp.

References

Erdős, Faudree, Rousseau and Schelp, Ramsey size linear graphs, Combin. Probab. Comput. 2 (1993), 389–399.

Imports (6)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.HasFiniteRamseyProperty The finite Ramsey property for the pair line 55 source
Source documentation
The finite Ramsey property for the pair $(G_1, G_2)$: for every $n \geq 1$, there exists a $G_1$-free graph H on some vertex type in Type (universe 0) such that every n-edge-colouring of H contains a monochromatic copy of G_2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: Nat, LE.le, OfNat.ofNat, Exists, And, SimpleGraph.Free, Fin, SimpleGraph.IsEdgeColouring, SimpleGraph.IsContained
2 instances, coercions and auxiliary constants left out.
```
def HasFiniteRamseyProperty {U₁ U₂ : Type*}
    (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂) : Prop :=
…
```
def SimpleGraph.HasCountableRamseyEscape The countable Ramsey escape property for the pair line 66 source
Source documentation
The countable Ramsey escape property for the pair $(G_1, G_2)$: every $G_1$-free graph H on a Type-valued vertex type has a countable edge-colouring in which every colour class is $G_2$-free.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: SimpleGraph.Free, Exists, Nat, And, SimpleGraph.IsEdgeColouring
```
def HasCountableRamseyEscape {U₁ U₂ : Type*}
    (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂) : Prop :=
…
```
def SimpleGraph.IsErdosHajnalExceptional A pair $(G_1, G_2)$ is Erdős–Hajnal exceptional if it has both the finite Ramsey property and the countable Ramsey escape property line 75 source
Source documentation
A pair $(G_1, G_2)$ is Erdős–Hajnal exceptional if it has both the finite Ramsey property and the countable Ramsey escape property.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: And, SimpleGraph.HasFiniteRamseyProperty, SimpleGraph.HasCountableRamseyEscape
```
def IsErdosHajnalExceptional {U₁ U₂ : Type*}
    (G₁ : SimpleGraph U₁) (G₂ : SimpleGraph U₂) : Prop :=
…
```
def SimpleGraph.graphRamsey Documented declaration: graphRamsey line 87 source
Source documentation
The two-color Ramsey number graphRamsey G H is the minimum number of vertices n such that every 2-coloring of the edges of the complete graph on n vertices contains a copy of G in the first color or a copy of H in the second color.
A 2-coloring of the complete graph on Fin n is represented by a graph C (the edges of the first color) and its complement Cᶜ (the edges of the second color).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Nat
Proof uses: InfSet.sInf, Set.ofPred, Fin, Or, SimpleGraph.IsContained, Compl.compl
2 instances, coercions and auxiliary constants left out.
```
noncomputable def graphRamsey {α β : Type*} [Fintype α] [Fintype β]
    (G : SimpleGraph α) (H : SimpleGraph β) : ℕ :=
…
```
def SimpleGraph.diagonalGraphRamsey The diagonal graph Ramsey number R(G, G) line 92 source
Source documentation
The diagonal graph Ramsey number R(G, G).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph, Nat
Proof uses: SimpleGraph.graphRamsey
```
noncomputable def diagonalGraphRamsey {α : Type*} [Fintype α] (G : SimpleGraph α) : ℕ :=
…
```
def SimpleGraph.classicalRamsey The classical two-color Ramsey number R(k, l) = R(K_k, K_l) line 96 source
Source documentation
The classical two-color Ramsey number R(k, l) = R(K_k, K_l).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SimpleGraph.graphRamsey, Fin, SimpleGraph.completeGraph
1 instances, coercions and auxiliary constants left out.
```
noncomputable def classicalRamsey (k l : ℕ) : ℕ :=
…
```
Mentions: SimpleGraph.l
def SimpleGraph.diagonalRamsey The diagonal classical Ramsey number R(k) = R(K_k, K_k) line 100 source
Source documentation
The diagonal classical Ramsey number R(k) = R(K_k, K_k).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SimpleGraph.classicalRamsey
```
noncomputable def diagonalRamsey (k : ℕ) : ℕ :=
…
```
def SimpleGraph.IsRamseySizeLinear Documented declaration: IsRamseySizeLinear line 111 source
Source documentation
A graph G is Ramsey size linear if there exists a constant c > 0 such that for all graphs H with m edges and no isolated vertices, the Ramsey number satisfies R(G, H) ≤ c · m.
Note that this is about the (vertex) Ramsey number graphRamsey G H, not the size Ramsey number sizeRamsey G H.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph
Proof uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Fin, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, LE.le, SimpleGraph.graphRamsey, HMul.hMul, Set.ncard, Sym2, SimpleGraph.edgeSet
14 instances, coercions and auxiliary constants left out.
```
def IsRamseySizeLinear {α : Type*} [Fintype α] (G : SimpleGraph α) : Prop :=
…
```