Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?

AMS 11 · solved (literature)

theorem erdos_1209.parts.i :
    answer(False) ↔
      ∃ f : ℕ → ℕ, ∀ a : ℕ → ℕ, StrictMono a → (∀ k, f k ≤ a k) →
        (∃ n, ∀ k, (n + a k).Prime) →
        {n | ∀ k, (n + a k).Prime}.Infinite