FormalConjectures.ErdosProblems.«600»

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import FormalConjecturesUtil

Erdős Problem 600

Reference:

erdosproblems.com/600

erdosproblems.com/80

[Er87] Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.

[RuSz78] Ruzsa, I. Z. and Szemerédi, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II (1978), 939-945.


open Classical Filter

open scoped Topology


namespace Erdos600

Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles.

private noncomputable def trianglesContaining
  {α : Type*}
  (G : SimpleGraph α)
  (uv : Sym2 α)
  [Fintype α] :
  Finset (Finset α) :=
  (G.cliqueFinset 3).filter (fun t ↦ uv.toFinset ⊆ t)


def Erdos600Prop (n : ℕ) (e : ℕ) (r : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin n), G.edgeFinset.card ≥ e →
  (∀ uv ∈ G.edgeFinset, (trianglesContaining G uv).Nonempty) →
  ∃ uv ∈ G.edgeFinset, r ≤ (trianglesContaining G uv).card


noncomputable def eFunction (n : ℕ) (r : ℕ) : ℕ := sInf {e : ℕ | Erdos600Prop n e r}

Let $r \geq 2$. Is it true that $e(n,r+1) - e(n,r) \to \infty$ as $n \to \infty$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_600.parts.i :
    answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →
      Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) - (eFunction n r : ℝ)) atTop atTop := by⊢ True ↔ ∀ (r : ℕ), 2 ≤ r → Tendsto (fun n => ↑(eFunction n (r + 1)) - ↑(eFunction n r)) atTop atTop
  sorryAll goals completed! 🐙

Let $r \geq 2$. Is it true that $\frac{e(n,r+1)}{e(n,r)} \to 1$ as $n \to \infty$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_600.parts.ii :
    answer(sorry) ↔ ∀ r : ℕ, 2 ≤ r →
      Tendsto (fun (n : ℕ) ↦ (eFunction n (r + 1) : ℝ) / (eFunction n r : ℝ)) atTop (𝓝 1) := by⊢ True ↔ ∀ (r : ℕ), 2 ≤ r → Tendsto (fun n => ↑(eFunction n (r + 1)) / ↑(eFunction n r)) atTop (𝓝 1)
  sorryAll goals completed! 🐙

Ruzsa and Szemerédi [RuSz78] proved that $e(n,r)=o(n^2)$ for any fixed $r$.

@[category research solved, AMS 5]
theorem declaration uses 'sorry'erdos_600.variants.ruzsa_szemeredi_upper_bound :
    ∀ r : ℕ, (fun (n : ℕ) ↦ (eFunction n r : ℝ)) =o[atTop] (fun (n : ℕ) ↦ (n : ℝ)^2) := by⊢ ∀ (r : ℕ), (fun n => ↑(eFunction n r)) =o[atTop] fun n => ↑n ^ 2
  sorryAll goals completed! 🐙


end Erdos600