  The problem of determining the maximum size $La(n,P)$ that a $P$-free
subposet of the Boolean lattice $B_n$ can have, attracted the attention of many
researchers, but little is known about the induced version of these problems.
In this paper we determine the asymptotic behavior of $La^*(n,P)$, the maximum
size that an induced $P$-free subposet of the Boolean lattice $B_n$ can have
for the case when $P$ is the complete two-level poset $K_{r,t}$ or the complete
multi-level poset $K_{r,s_1,\dots,s_j,t}$ when all $s_i$'s either equal 4 or
are large enough and satisfy an extra condition. We also show lower and upper
bounds for the non-induced problem in the case when $P$ is the complete
three-level poset $K_{r,s,t}$. These bounds determine the asymptotics of
$La(n,K_{r,s,t})$ for some values of $s$ independently of the values of $r$ and
$t$.
