We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the minimum degree.Let $ϵ,\alpha >0$ and $D$ and $n$ be sufficiently large. If $G$ is a graph on $n$ vertices with at least $n^{1+\alpha }$ edges, then must $G$ contain a $D$-balanced subgraph on $m>n^{1-\alpha }$ vertices with at least $ϵm^{1+\alpha }$ edges?

AMS 5 · solved (literature)

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)