Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$
are prime for all $a\in A$ with $0 < a < n$ and \[\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?\]
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theorem erdos_428 :
answer(sorry) ↔ ∃ A : Set ℕ,
(∃ᶠ n in atTop, ∀ a ∈ A, 0 < a → a < n → (n - a).Prime) ∧
liminf (fun n ↦ primeDensityRatio A n) atTop > 0
