594.lean`

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3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 594

References:

erdosproblems.com/594
[ErHa66] Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.
[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
[EHS74] Erdős, P. and Hajnal, A. and Shelah, S., On some general properties of chromatic numbers. Topics in topology (Proc. Colloq., Keszthely, 1972) (1974), 243-255.

Imports (1)

FormalConjecturesUtil

theorem Erdos594.erdos_594 Erdős Problem 594 (Erdős–Hajnal [ErHa66], [Er69b]): Does every graph $G$ with chromatic number $\geq \aleph_1$ contain all sufficiently large odd cycles? sorry line 55 source
Source documentation
Erdős Problem 594 (Erdős–Hajnal [ErHa66], [Er69b]):
Does every graph $G$ with chromatic number $\geq \aleph_1$ contain all sufficiently large odd cycles?
The answer is Yes, proved by Erdős, Hajnal, and Shelah [EHS74].
A graph has chromatic number $\geq \aleph_1$ (i.e. uncountable chromatic number) if and only if it admits no proper colouring with countably many colours; this is encoded as IsEmpty (G.Coloring ℕ). The conclusion states that there is some $N$ such that for every $k \geq N$ the graph contains a cycle of odd length $2k + 1$.

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph, IsEmpty, SimpleGraph.Coloring, Nat, Exists, LE.le, SimpleGraph.Walk, And, SimpleGraph.Walk.IsCycle, Eq, SimpleGraph.Walk.length, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_594 : answer(True) ↔
    ∀ (V : Type) (G : SimpleGraph V), IsEmpty (G.Coloring ℕ) →
      ∃ N : ℕ, ∀ k : ℕ, N ≤ k →
        ∃ (v : V) (w : G.Walk v v), w.IsCycle ∧ w.length = 2 * k + 1 := by
…
```
theorem Erdos594.erdos_594.variants.erdos_hajnal The earlier result of Erdős and Hajnal [ErHa66]: every graph with chromatic number $\geq \aleph_2$ contains all sufficiently large odd cycles sorry line 69 source
Source documentation
The earlier result of Erdős and Hajnal [ErHa66]: every graph with chromatic number $\geq \aleph_2$ contains all sufficiently large odd cycles.
Chromatic number $\geq \aleph_2$ is encoded as the nonexistence of a proper colouring with any set of at most $\aleph_1$ colours.

depends on sorry what this means
Statement uses: SimpleGraph, LE.le, Cardinal, Cardinal.mk, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, IsEmpty, SimpleGraph.Coloring, Exists, Nat, SimpleGraph.Walk, And, SimpleGraph.Walk.IsCycle, Eq, SimpleGraph.Walk.length, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_594.variants.erdos_hajnal (V : Type) (G : SimpleGraph V)
    (h : ∀ (α : Type), #α ≤ ℵ_ 1 → IsEmpty (G.Coloring α)) :
    ∃ N : ℕ, ∀ k : ℕ, N ≤ k →
      ∃ (v : V) (w : G.Walk v v), w.IsCycle ∧ w.length = 2 * k + 1 := by
…
```
theorem Erdos594.erdos_594.variants.bot_countably_colorable Sanity check for the encoding of "chromatic number line 81 source
Source documentation
Sanity check for the encoding of "chromatic number $\geq \aleph_1$": a countably colourable graph, such as the empty graph on $\mathbb{N}$, does not satisfy the hypothesis IsEmpty (G.Coloring ℕ).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, IsEmpty, SimpleGraph.Coloring, Nat, Bot.bot, SimpleGraph
Proof uses: Eq.mpr, Nonempty, id, Eq, congrArg, propext, not_isEmpty_iff, Nonempty.intro, SimpleGraph.Coloring.mk, OfNat.ofNat, of_eq_true, SimpleGraph.Adj, Eq.trans, True, forall_congr, False, implies_congr, eq_self, not_true_eq_false, implies_true
11 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_594.variants.bot_countably_colorable :
    ¬ IsEmpty ((⊥ : SimpleGraph ℕ).Coloring ℕ) := by
…
```