FormalConjectures.ErdosProblems.«572»

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module

public import FormalConjecturesUtil

Erdős Problem 572

References:

erdosproblems.com/572

[Er64c] Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (1964), 29-36.

[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combin. Theory Ser. B (1974), 97-105.

[LUW95] Lazebnik, F., Ustimenko, V. A. and Woldar, A. J., A new series of dense graphs of high girth. Bull. Amer. Math. Soc. (N.S.) (1995), 73-79.

@[expose] public sectionopen Filternamespace Erdos572

Show that for $k\geq 3$ $$\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.$$

This problem is #46 in Extremal Graph Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_572 (k : ℕ) (hk : 3 ≤ k) :
    ∃ c > (0 : ℝ), ∀ᶠ (n : ℕ) in atTop,
      c * (n : ℝ) ^ (1 + 1 / (k : ℝ)) ≤
        (SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph (2 * k)) : ℝ) := byk:ℕhk:3 ≤ k⊢ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, c * ↑n ^ (1 + 1 / ↑k) ≤ ↑(SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph (2 * k)))
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos572