Let $Q =(q_i)$ be a finite set of prime numbers in $\Bbb{Z}$. Define $\hat{Q}$ to be the unique largest set of primes reachable by adding an offset to $Q$:
$$ \hat{Q} = \bigcup_{a \in \Bbb{Z} \\ Q + a \text{ all prime}} (Q + a) $$
where $Q + a := Q + \{q\} = \{ q + a: q \in Q \}$.
Then is $\hat{Q}$ always finite when $|Q| \gt 1$?
Suppose it's true for all $|Q| \lt n$. Then when $|Q| = n$ we have $Q \subset Q \cup p\subset \widehat{Q \cup p} \subset \widehat{Q} \cap \hat{p}$ ??
When $Q$ is any set of twin primes, e.g. $Q = \{3,5\}$, then $\hat{Q}$ will contain all twin primes, and we think there's lots of those.
More generally, we know that for some not-terribly-large integer $N$, there are infinitely many primes that differ by $N$. Put two such primes in $Q$, and you'll get back all of them in $\hat{Q}$.