FormalConjectures.ErdosProblems.«740»

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import FormalConjecturesUtil

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

Reference: erdosproblems.com/740


open scoped Cardinal


namespace Erdos740

A graph avoids odd cycles of length $\leq r$ if it contains no odd cycles of length at most $r$.

def NoShortOddCycle {V : Type*} (G : SimpleGraph V) (r : ℕ) : Prop :=
  ∀ (v : V) (c : G.Walk v v), c.IsCycle → Odd c.length → c.length > r

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 $\leq r$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'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⊢ True ↔
  ∀ (V : Type u_1) (G : SimpleGraph V),
    ℵ₀ ≤ G.chromaticCardinal → ∀ (r : ℕ), ∃ H, H.coe.chromaticCardinal = G.chromaticCardinal ∧ NoShortOddCycle H.coe r
  sorryAll goals completed! 🐙


-- TODO: add the related infinitary chromatic-number statements from erdosproblems.com.

end Erdos740