FormalConjectures.ErdosProblems.«75»

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import FormalConjecturesUtil

Erdős Problem 75

Reference:

[erdosproblems.com/75] (https://www.erdosproblems.com/75)


open Cardinal


namespace Erdos75

Is there a graph of chromatic number ℵ_ 1 with ℵ_ 1 vertices such that for all ε > 0, if n is sufficiently large and H is a subgraph on n vertices, then H contains an independent set of size > n ^ (1 - ε)?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_75 :
    answer(sorry) ↔
    ∃ (V : Type) (G : SimpleGraph V),
      G.chromaticCardinal = ℵ_ 1 ∧
      #V = ℵ_ 1 ∧
      ∀ ε > (0 : ℝ),
        ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (H : G.Subgraph),
            H.verts.ncard = n →
            ∃ (I : Finset V),
              (I : Set V) ⊆ H.verts ∧
              G.IsIndepSet (I : Set V) ∧
              (I.card : ℝ) > (n : ℝ) ^ (1 - ε) := by⊢ True ↔
  ∃ V G,
    G.chromaticCardinal = ℵ_ 1 ∧
      #V = ℵ_ 1 ∧
        ∀ ε > 0,
          ∀ᶠ (n : ℕ) in Filter.atTop,
            ∀ (H : G.Subgraph), H.verts.ncard = n → ∃ I, ↑I ⊆ H.verts ∧ G.IsIndepSet ↑I ∧ ↑I.card > ↑n ^ (1 - ε)
  sorryAll goals completed! 🐙


end Erdos75