EdgeColouring.lean`

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

2 declarations, 0 contain sorry, 0 declared as axioms

Edge colourings of a SimpleGraph

SimpleGraph.IsEdgeColouring H c asserts that c : ι → SimpleGraph V is a partition of the edges of H into colour classes indexed by ι. The classes must (1) cover H (H = ⨆ i, c i) and (2) be pairwise disjoint (each edge belongs to exactly one class).

This unifies the previous local IsNEdgeColouring (ι = Fin n) and IsCountableEdgeColouring (ι = ℕ) shapes used in FormalConjectures/ErdosProblems/596.lean. The two are now special cases of IsEdgeColouring parameterised by the index type.

SimpleGraph.TopEdgeLabeling.CliqueFree is the Ramsey-theoretic condition on the other edge-colouring shape Mathlib provides, a labelling of the edges of the complete graph: no colour class contains a clique of the given size.

Imports (3)

Imported by 2 files

def SimpleGraph.IsEdgeColouring IsEdgeColouring H c asserts that c : ι → SimpleGraph V is a partition of the edges of H: the colour classes cover H (H = ⨆ i, c i) and are pairwise disjoint line 44 source
Source documentation
IsEdgeColouring H c asserts that c : ι → SimpleGraph V is a partition of the edges of H: the colour classes cover H (H = ⨆ i, c i) and are pairwise disjoint.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: And, Eq, iSup, Ne, Disjoint
8 instances, coercions and auxiliary constants left out.
```
def IsEdgeColouring {V ι : Type*} (H : SimpleGraph V) (c : ι → SimpleGraph V) : Prop :=
…
```
Mentions: c
def SimpleGraph.TopEdgeLabeling.CliqueFree A labelling of the edges of the complete graph on V by ι is n-clique-free when no colour class contains an n-clique, that is, when it admits no monochromatic n-clique line 49 source
Source documentation
A labelling of the edges of the complete graph on V by ι is n-clique-free when no colour class contains an n-clique, that is, when it admits no monochromatic n-clique.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph.TopEdgeLabeling, Nat
Proof uses: SimpleGraph.CliqueFree, SimpleGraph.EdgeLabeling.labelGraph, Top.top, SimpleGraph
4 instances, coercions and auxiliary constants left out.
```
def TopEdgeLabeling.CliqueFree {V ι : Type*} (C : TopEdgeLabeling V ι) (n : ℕ) : Prop :=
…
```