  Let $(a_1,\dots, a_m)$ be an $m$-tuple of positive, pairwise distinct,
integers. If for all $1\leq i< j \leq m$ the prime divisors of $a_ia_j+1$ come
from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In
this note we estimate the number of $S$-Diophantine quadruples in terms of
$|S|=r$.
