FormalConjectures.ErdosProblems.«713»

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module

public import FormalConjecturesUtil

Erdős Problem 713

References:

erdosproblems.com/713

[Er67d] Erdős, P., Some recent results on extremal problems in graph theory. {R}esults. (1967), 117--123 (English); pp. 124--130 (French).

[ErSi70] Erdős, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonfüred, 1969) (1970), 377-390.

[FrFu87] Frankl, P. and Füredi, Z., Exact solution of some Turán-type problems. J. Combin. Theory Ser. A (1987), 226--262.

[FuGe21] Füredi, Zoltán and Gerbner, Dániel, Hypergraphs without exponents. J. Combin. Theory Ser. A (2021), Paper No. 105517, 9.

@[expose] public sectionopen Filter SimpleGraphnamespace Erdos713open scoped Classical in

Is it true that, for every bipartite graph $G$, there exists some $\alpha\in [1,2)$ and $c>0$ such that $$\mathrm{ex}(n;G)\sim cn^\alpha?$$

The condition that $G$ have at least two edges excludes degenerate forbidden graphs whose extremal number is eventually zero, for which the displayed asymptotic with $c>0$ is impossible.

@[category research open, AMS 5]
theorem erdos_713.parts.i : answer(sorry) ↔
    ∀ (q : ℕ) (G : SimpleGraph (Fin q)), G.IsBipartite → 2 ≤ G.edgeFinset.card →
      ∃ α c : ℝ, α ∈ Set.Ico 1 2 ∧ 0 < c ∧
        Asymptotics.IsEquivalent atTop
          (fun n : ℕ => (extremalNumber n G : ℝ))
          (fun n : ℕ => c * (n : ℝ) ^ α) := by⊢ True ↔
  ∀ (q : ℕ) (G : SimpleGraph (Fin q)),
    G.IsBipartite →
      2 ≤ G.edgeFinset.card →
        ∃ α c,
          α ∈ Set.Ico 1 2 ∧ 0 < c ∧ Asymptotics.IsEquivalent atTop (fun n ↦ ↑(extremalNumber n G)) fun n ↦ c * ↑n ^ α
  sorryAll goals completed! 🐙

open scoped Classical in

Must $\alpha$ be rational?

The same nondegeneracy condition on $G$ is used as in part (i). Rationality means that the real number $\alpha$ lies in the image of the canonical embedding $\mathbb{Q}\to\mathbb{R}$.

@[category research open, AMS 5]
theorem erdos_713.parts.ii : answer(sorry) ↔
    ∀ (q : ℕ) (G : SimpleGraph (Fin q)), G.IsBipartite → 2 ≤ G.edgeFinset.card →
      ∀ α c : ℝ, α ∈ Set.Ico 1 2 → 0 < c →
        Asymptotics.IsEquivalent atTop
          (fun n : ℕ => (extremalNumber n G : ℝ))
          (fun n : ℕ => c * (n : ℝ) ^ α) →
        α ∈ Set.range ((↑) : ℚ → ℝ) := by⊢ True ↔
  ∀ (q : ℕ) (G : SimpleGraph (Fin q)),
    G.IsBipartite →
      2 ≤ G.edgeFinset.card →
        ∀ (α c : ℝ),
          α ∈ Set.Ico 1 2 →
            0 < c →
              (Asymptotics.IsEquivalent atTop (fun n ↦ ↑(extremalNumber n G)) fun n ↦ c * ↑n ^ α) →
                α ∈ Set.range Rat.cast
  sorryAll goals completed! 🐙

end Erdos713