  The independent set reconfiguration problem asks whether one can transform
one given independent set of a graph into another, by changing vertices one by
one in such a way the intermediate sets remain independent. Extremal problems
on independent sets are widely studied: for example, it is well known that an
$n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is
tight). This paper investigates the asymptotic behavior of maximum possible
length of a shortest reconfiguration sequence for independent sets of size $k$
among all $n$-vertex graphs. We give a tight bound for $k=2$. We also provide a
subquadratic upper bound (using the hypergraph removal lemma) as well as an
almost tight construction for $k=3$. We generalize our results for larger
values of $k$ by proving an $n^{2\lfloor k/3 \rfloor}$ lower bound.
