A note on the number of $S$-Diophantine quadruples

Let $(a_1,\dots ,a_m)$ be an $m$-tuple of positive, pairwise distinct, integers. If for all $1\le i<j\le m$ the prime divisors of $a_i{a}_j+1$ 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 $\mathrm{\mid }S\mathrm{\mid }=r$.