  In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets $F_1,F_2, \dots,F_{|P|}$ form a
copy of a poset $P$, if there exists a bijection $i:P\rightarrow \{F_1,F_2,
\dots,F_{|P|}\}$ such that for any $p,p'\in P$ the relation $p<_P p'$ implies
$i(p)\subsetneq i(p')$. A family $\mathcal{F}$ of sets is \textit{$P$-free} if
it does not contain any copy of $P$. The trace of a family $\mathcal{F}$ on a
set $X$ is $\mathcal{F}|_X:=\{F\cap X: F\in \mathcal{F}\}$.
  We introduce the following notions: $\mathcal{F}\subseteq 2^{[n]}$ is
$l$-trace $P$-free if for any $l$-subset $L\subseteq [n]$, the family
$\mathcal{F}|_L$ is $P$-free and $\mathcal{F}$ is trace $P$-free if it is
$l$-trace $P$-free for all $l\le n$. As the first instances of these problems
we determine the maximum size of trace $B$-free families, where $B$ is the
butterfly poset on four elements $a,b,c,d$ with $a,b<c,d$ and determine the
asymptotics of the maximum size of $(n-i)$-trace $K_{r,s}$-free families for
$i=1,2$. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems.
