Knowing that $p$ is prime enables us to rule out the possibility that $p+2$ and $p+4$ are both prime, except in the one trivial case that $p=3$, since at least one of $p,\ p+2,\ p+4$ is divisible by $3$. But in some cases, $p,\ p+2,\ p+6$ are all prime.
For which finite sets $0\in A\subseteq \{0,2,4,6,\ldots\}$ does there exist a prime $p$ such that every member of $p+A = \{p+a : a\in A\}$ is prime?
Are there some such sets $A$ (besides $A=\lbrace 0 \rbrace$) for which infinitely many such $p$ are known to exist? (I think the answer to that one is unknown in the case $A=\{0,2\}$.)
Later note: Above I wrote $A\subseteq \{0,2,4,6,\ldots\}$. Later I changed it to $0\in A\subseteq \{0,2,4,6,\ldots\}$. Any $A$ that doesn't contain $0$ represents the same size and shape as one that does.