740.lean`

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 740: Infinitary version of chromatic number and odd cycles

Reference: erdosproblems.com/740

Imports (1)

FormalConjecturesUtil

def Erdos740.NoShortOddCycle A graph avoids odd cycles of length $\leq r$ if it contains no odd cycles of length at most $r$ line 33 source
Source documentation
A graph avoids odd cycles of length $\leq r$ if it contains no odd cycles of length at most $r$.

standard axioms: propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: SimpleGraph.Walk, SimpleGraph.Walk.IsCycle, Odd, SimpleGraph.Walk.length, GT.gt
2 instances, coercions and auxiliary constants left out.
```
def NoShortOddCycle {V : Type*} (G : SimpleGraph V) (r : ℕ) : Prop :=
…
```
theorem Erdos740.erdos_740 Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$ sorry line 42 source
Source documentation
Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $r\geq 1$. Must $G$ contain a subgraph of chromatic number $\mathfrak{m}$ which does not contain any odd cycle of length $\leq r$?

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph, LE.le, Cardinal, Cardinal.aleph0, SimpleGraph.chromaticCardinal, Nat, Exists, SimpleGraph.Subgraph, And, Eq, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, Erdos740.NoShortOddCycle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_740 :
    answer(sorry) ↔
      ∀ (V : Type*) (G : SimpleGraph V),
        ℵ₀ ≤ G.chromaticCardinal →
          ∀ (r : ℕ),
            ∃ (H : G.Subgraph), H.coe.chromaticCardinal = G.chromaticCardinal ∧
              NoShortOddCycle H.coe r := by
…
```
Mentions: Erdos740.NoShortOddCycle