FormalConjectures.ErdosProblems.«765»

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import FormalConjecturesUtil

Erdős Problem 765

References:

erdosproblems.com/765

[Er38] Erdős, P., On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.

[Re58] Reiman, I., Über ein Problem von K. Zarankiewicz. Acta Math. Acad. Sci. Hungar. 9 (1958), 269-273.

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

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

[Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14.

[Fu83] Füredi, Z., Graphs without quadrilaterals. J. Combin. Theory Ser. B 34 (1983), 187-190.

[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.

[MaYa23] Ma, Jie and Yang, Tianchi, Upper bounds on the extremal number of the 4-cycle. Bull. Lond. Math. Soc. (2023), 1655--1667.

open Filter Asymptoticsnamespace Erdos765

Give an asymptotic formula for $\mathrm{ex}(n; C_4)$.

Erdős and Klein [Er38] proved $\mathrm{ex}(n; C_4) \asymp n^{3/2}$, and Reiman [Re58] proved $$\frac{1}{2\sqrt 2} \le \lim \frac{\mathrm{ex}(n; C_4)}{n^{3/2}} \le \frac12.$$ Erdős and Rényi [ERS66], and independently Brown [Br66], gave a construction showing that if $n = q^2 + q + 1$ with $q$ a prime power then $\mathrm{ex}(n; C_4) \ge \frac12 q (q+1)^2$; together with Reiman's upper bound this gives $\mathrm{ex}(n; C_4) \sim \frac12 n^{3/2}$. Füredi [Fu83] proved $\mathrm{ex}(n; C_4) = \frac12 q (q+1)^2$ for $q > 13$.

@[category research solved, AMS 5, formal_proof using lean4 at
  "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos765.lean#L24"]
theorem erdos_765 :
    (fun n : ℕ ↦ (SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4) : ℝ)) ~[atTop]
      fun n : ℕ ↦ (n : ℝ) ^ (3 / 2 : ℝ) / 2 := by⊢ (fun n ↦ ↑(SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4))) ~[atTop] fun n ↦ ↑n ^ (3 / 2) / 2
  sorryAll goals completed! 🐙

Erdős [Er93] conjectured that $\mathrm{ex}(n; C_4) = \frac{n^{3/2}}{2} + \frac n4 + O(n^{1/2})$ for all $n$, having proved the upper bound $\mathrm{ex}(n; C_4) \le \frac{n^{3/2}}{2} + \frac n4 + O(n^{1/2})$ in [Er75]. This is false: Ma and Yang [MaYa23] proved that, for some absolute constant $c > 0$ and a positive density set of $n$, $\mathrm{ex}(n; C_4) \le \frac{n^{3/2}}{2} + (\frac14 - c) n$.

@[category research solved, AMS 5]
theorem erdos_765.variants.second_term : answer(False) ↔
    (fun n : ℕ ↦ (SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4) : ℝ) -
      (n : ℝ) ^ (3 / 2 : ℝ) / 2 - (n : ℝ) / 4) =O[atTop] fun n : ℕ ↦ (n : ℝ) ^ (1 / 2 : ℝ) := by⊢ False ↔
  (fun n ↦ ↑(SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph 4)) - ↑n ^ (3 / 2) / 2 - ↑n / 4) =O[atTop] fun n ↦
    ↑n ^ (1 / 2)
  sorryAll goals completed! 🐙

end Erdos765