It is known that for any two positive coprime integers $a$ and $d$, there are infinitely many primes of the form $a + nd$, where $n$ is a non-negative integer. This is known as the Dirichlet's theorem on arithmetic progressions.
So I am wondering, given $k+1$ different prime numbers: $p_0,p_1,...,p_k$, with $p_0=2$ and $k>0$, is it true that there are infinitely many primes $P$ such that all prime divisors of $P-1$ are only $p_0,p_1,...,p_k$ ? In other words, is it true that there are infinitely many primes $P$ of the form $p_0^{\alpha_0}p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}+1$ ?
(Please let me know if this question is off-topic or should be closed)