FormalConjectures.ErdosProblems.«714»

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module

public import FormalConjecturesUtil

Erdős Problem 714

References:

erdosproblems.com/714

[Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.

[ERS66] Erdős, P. and Rényi, A. and Sós, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.

[KST54] Kövari, T. and Sós, V. T. and Turán, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.

@[expose] public sectionopen Filter SimpleGraphnamespace Erdos714

Is it true that $$\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?$$

@[category research open, AMS 5]
theorem erdos_714 : answer(sorry) ↔
    ∀ r : ℕ, 2 ≤ r → ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
      c * (n : ℝ) ^ ((2 : ℝ) - 1 / (r : ℝ)) ≤
        (extremalNumber n (completeBipartiteGraph (Fin r) (Fin r)) : ℝ) := by⊢ True ↔
  ∀ (r : ℕ),
    2 ≤ r →
      ∃ c,
        0 < c ∧
          ∀ᶠ (n : ℕ) in atTop, c * ↑n ^ (2 - 1 / ↑r) ≤ ↑(extremalNumber n (completeBipartiteGraph (Fin r) (Fin r)))
  sorryAll goals completed! 🐙

end Erdos714