Here's the basic idea. For 3 consecutive numbers, one can look at $x^m-1$, $x^m$, $x^m+1$, where $m$ is odd and has many small prime factors. Certainly $x^m$ has only small prime factors relative to its size. And $x^m-1$ and $x^m+1$ factor into cyclotomic polynomials of degree at most $\phi(m)$. So all the prime factors should be of size at most about $x^{\phi(m)}=(x^m)^{\phi(m)/m}$, and $\phi(m)/m$ can be made as small as one wants (by pumping $m$ full of small odd primes).