What is the greatest constant $c$ such that for any primes $p, q$ with $p<q<cp$, there exist two consecutive positive integers, one with largest prime divisor $p$, and the other with largest prime divisor $q$?
This question shows that $c=2$ works, and the argument uses exactly $c=2$. What can go wrong if $c>2$? For many pairs of primes, it looks like finding such consecutive integers is still possible. For example if $p=3$ and $q=29$, we can take $(144,145)$.