Ramsey.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

Hypergraph Ramsey Numbers

This file defines the r-uniform hypergraph Ramsey number R_r(n).

Imports (3)

Imported by 2 files

def Combinatorics.hypergraphRamsey The r-uniform hypergraph Ramsey number R_r(n) line 37 source
Source documentation
The r-uniform hypergraph Ramsey number R_r(n). The smallest natural number m such that every 2-coloring of the r-subsets of a set of size m contains a monochromatic subset of size n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Finset, Fin, Bool, Exists, And, Eq, Finset.card, LE.le
4 instances, coercions and auxiliary constants left out.
```
noncomputable def hypergraphRamsey (r n : ℕ) : ℕ :=
…
```
theorem Combinatorics.le_hypergraphRamsey n ≤ hypergraphRamsey r n when the set is nonempty line 43 source
Source documentation
n ≤ hypergraphRamsey r n when the set is nonempty.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set.Nonempty, Set.ofPred, Finset, Fin, Bool, Exists, And, Eq, Finset.card, LE.le, Combinatorics.hypergraphRamsey
Proof uses: le_csInf, Membership.mem, Set, Bool.false, Trans.trans, Fintype.card, Eq.symm, Finset.card_le_univ, Fintype.card_fin
20 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem le_hypergraphRamsey (r n : ℕ) (hne : { m | ∀ (c : Finset (Fin m) → Bool),
    ∃ (S : Finset (Fin m)), S.card = n ∧
      ∃ (color : Bool), ∀ (e : Finset (Fin m)), e ⊆ S → e.card = r → c e = color }.Nonempty) :
    n ≤ hypergraphRamsey r n := by
…
```
Mentions: c, Combinatorics.hypergraphRamsey
theorem Combinatorics.hypergraphRamsey_self hypergraphRamsey r r = r for any r line 56 source
Source documentation
hypergraphRamsey r r = r for any r.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Combinatorics.hypergraphRamsey
Proof uses: Membership.mem, Set, Set.ofPred, Finset, Fin, Bool, Exists, And, Finset.card, LE.le, Exists.intro, Finset.univ, And.intro, of_eq_true, Eq.trans, True, congrFun', congrArg, Fintype.card_fin, eq_self and 8 more
6 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem hypergraphRamsey_self (r : ℕ) : hypergraphRamsey r r = r := by
…
```
Mentions: Combinatorics.hypergraphRamsey