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module
public import Mathlib.Combinatorics.SimpleGraph.Copy
public import Mathlib.Data.Real.Basic
public import Mathlib.Data.Set.Card
public import Mathlib.Order.Lattice.Nat@[expose] public sectionSize Ramsey Number
This file defines the size Ramsey number for simple graphs.
Definition
The size Ramsey number r̂(G, H) is the minimum number of edges in a graph F
such that any 2-coloring of F's edges contains a copy of G in one color or H in the other.
namespace SimpleGraph
The size Ramsey number r̂(G, H) is the minimum number of edges in a graph F
such that any 2-coloring of F's edges contains a copy of G in one color or H in the other.
A 2-coloring is represented by a subgraph R ≤ F (the "red" edges); the "blue" edges are F \ R.
noncomputable def sizeRamsey {α β : Type*} [Fintype α] [Fintype β]
(G : SimpleGraph α) (H : SimpleGraph β) : ℕ :=
sInf { m | ∃ (n : ℕ) (F : SimpleGraph (Fin n)),
F.edgeSet.ncard = m ∧
∀ (R : SimpleGraph (Fin n)), R ≤ F →
G.IsContained R ∨ H.IsContained (F \ R) }
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, r̂(G, H) ≤ c · m.
def IsRamseySizeLinear {α : Type*} [Fintype α] (G : SimpleGraph α) : Prop :=
∃ c > (0 : ℝ), ∀ (n : ℕ) (H : SimpleGraph (Fin n)) [DecidableRel H.Adj],
(∀ v, 0 < H.degree v) →
(sizeRamsey G H : ℝ) ≤ c * H.edgeSet.ncardend SimpleGraph