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import FormalConjecturesUtilErdős Problem 1077
open Classical Finset Filter SimpleGraph
namespace Erdos1077
We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the minimum degree.
Let $ε, α > 0$ and $D$ and $n$ be sufficiently large. If $G$ is a graph on $n$ vertices with at least $n^{1+α}$ edges, then must $G$ contain a $D$-balanced subgraph on $m > n^{1-α}$ vertices with at least $εm^{1+α}$ edges?
@[category research solved, AMS 5]
theorem erdos_1077 :
answer(False) ↔ ∀ ε > (0 : ℝ), ε < 1 → ∀ α > (0 : ℝ), α < 1 → ∀ᶠ D in atTop, ∀ᶠ n in atTop,
∀ G : SimpleGraph (Fin n), G.edgeSet.ncard > (n : ℝ) ^ (1 + α) →
∃ (H : Subgraph G),
letI m := H.verts.ncard
IsBalanced H.coe D ∧
m > (n : ℝ) ^ (1 - α) ∧
H.edgeSet.ncard > ε * m ^ (1 + α) := ⊢ False ↔
∀ ε > 0,
ε < 1 →
∀ α > 0,
α < 1 →
∀ᶠ (D : ℝ) in atTop,
∀ᶠ (n : ℕ) in atTop,
∀ (G : SimpleGraph (Fin n)),
↑G.edgeSet.ncard > ↑n ^ (1 + α) →
∃ H,
H.coe.IsBalanced D ∧ ↑H.verts.ncard > ↑n ^ (1 - α) ∧ ↑H.edgeSet.ncard > ε * ↑H.verts.ncard ^ (1 + α)
All goals completed! 🐙
end Erdos1077