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import FormalConjecturesUtilErdős Problem 60
References:
[HeMaYa21] He, J. and Ma, J. and Yang, T., Some extremal results on 4-cycles. Journal of Combinatorial Theory B (2021).
namespace Erdos60open SimpleGraph Filteropen scoped RealDoes every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?
@[category research open, AMS 5]
theorem erdos_60 :
∃ c : ℝ, c > 0 ∧
∀ᶠ n : ℕ in atTop,
∀ (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
(extremalNumber n (cycleGraph 4) < G.edgeFinset.card) →
(c * Real.sqrt (n : ℝ) ≤ ({ H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard : ℝ)) := ⊢ ∃ c > 0,
∀ᶠ (n : ℕ) in atTop,
∀ (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj],
extremalNumber n (cycleGraph 4) < G.edgeFinset.card → c * √↑n ≤ ↑{H' | Nonempty (H'.coe ≃g cycleGraph 4)}.ncard
All goals completed! 🐙He, Ma, and Yang [HeMaYa21] proved the conjecture when $n = q^2 + q + 1$ for some even integer $q$.
@[category research solved, AMS 5]
theorem erdos_60.variants.he_ma_yang :
∃ c : ℝ, c > 0 ∧
∀ (q : ℕ) (_hq : Even q),
∀ (G : SimpleGraph (Fin (q^2 + q + 1))) [DecidableRel G.Adj],
(extremalNumber (q^2 + q + 1) (cycleGraph 4) < G.edgeFinset.card) →
(c * Real.sqrt ((q^2 + q + 1) : ℝ) ≤ ({ H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard : ℝ)) := ⊢ ∃ c > 0,
∀ (q : ℕ),
Even q →
∀ (G : SimpleGraph (Fin (q ^ 2 + q + 1))) [inst : DecidableRel G.Adj],
extremalNumber (q ^ 2 + q + 1) (cycleGraph 4) < G.edgeFinset.card →
c * √(↑q ^ 2 + ↑q + 1) ≤ ↑{H' | Nonempty (H'.coe ≃g cycleGraph 4)}.ncard
All goals completed! 🐙Erdős and Simonovits conjectured that at least 2 copies of $C_4$ are guaranteed.
@[category research solved, AMS 5]
theorem erdos_60.variants.two_copies :
∀ᶠ n : ℕ in atTop,
∀ (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
(extremalNumber n (cycleGraph 4) < G.edgeFinset.card) →
(2 ≤ { H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard) := ⊢ ∀ᶠ (n : ℕ) in atTop,
∀ (G : SimpleGraph (Fin n)) [inst : DecidableRel G.Adj],
extremalNumber n (cycleGraph 4) < G.edgeFinset.card → 2 ≤ {H' | Nonempty (H'.coe ≃g cycleGraph 4)}.ncard
All goals completed! 🐙end Erdos60