In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets F1,F2,…,F∣P∣ form a
copy of a poset P, if there exists a bijection i:P→{F1,F2,…,F∣P∣} such that for any p,p′∈P the relation p<Pp′ implies
i(p)⊊i(p′). A family F of sets is \textit{P-free} if
it does not contain any copy of P. The trace of a family F on a
set X is F∣X:={F∩X:F∈F}.
We introduce the following notions: F⊆2[n] is
l-trace P-free if for any l-subset L⊆[n], the family
F∣L is P-free and F is trace P-free if it is
l-trace P-free for all l≤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 Kr,s-free families for
i=1,2. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems