Erdős and Graham's sparse admissible-set conjecture

Let $A\subseteq\mathbb{N}$ be admissible if there is no prime $p$ such that $A$ contains at least one element in every residue class modulo $p$. Erdős and Graham's conjecture. There is a non-decreasing, unbounded function $f:\mathbb{N}\to\mathbb{Z}_{\geq 0}$ such that, whenever $A\subseteq\mathbb{N}$ is admissible and

$$|A\cap\{1,\dots,N\}|\leq f(N)$$

for all $N$, there exists $n\in\mathbb{Z}$ such that $A+n$ is contained in the positive primes. Erdős and Graham asked whether sufficiently sparse infinite admissible sets must have a translate contained in the primes. The conjecture is false: the paper constructs arbitrarily sparse infinite admissible sets with no such translate.