OPEN

Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)$ count the number of distinct possible values for $s_t(m)$ for $m\in [n+1,n+t]$. Is there an $\epsilon>0$ such that
\[f(n,t)>\epsilon t\]
for all $t$ and $n$?

Erdős and Graham report they can show
\[f(n,t) \gg \frac{t}{\log t}.\]