Let $C>0$ and $ϵ>0$ be sufficiently small. Are there infinitely many integers $a,b,n$ with $a\ge ϵn$ and $b\ge ϵn$ such that$$a!b!\mid n!(a+b-n)!$$and $a+b>n+C\log n$?

AMS 11 · solved (literature)

theorem erdos_728 :
    answer(True) ↔
      ∀ᶠ ε : ℝ in 𝓝[>] 0, ∀ C > (0 : ℝ), ∀ C' > C,
        ∃ a b n : ℕ,
          0 < n ∧
          ε * n < a ∧
          ε * n < b ∧
          a ! * b ! ∣ n ! * (a + b - n)! ∧
          a + b > n + C * log n ∧
          a + b < n + C' * log n