FormalConjectures.ErdosProblems.«600»

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

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 Filteropen scoped Topologynamespace Erdos600open scoped Classical in

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.

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