Generalized forbidden subposet problems

A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq {\cal F}$ of sets is a copy of a poset $P$ in ${\cal F}$ if there exists a bijection $\phi:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that whenever $x \le_P x'$ holds, then so does $\phi(x)\subseteq \phi(x')$. For a family ${\cal F}$ of sets, let $c(P,{\cal F})$ denote the number of copies of $P$ in ${\cal F}$, and we say that ${\cal F}$ is $P$-free if $c(P,{\cal F})=0$ holds. For any two posets $P,Q$ let us denote by $La(n,P,Q)$ the maximum number of copies of $Q$ over all $P$-free families ${\cal F} \subseteq 2^{[n]}$, i.e. $\max\{c(Q,{\cal F}): {\cal F} \subseteq 2^{[n]}, c(P,{\cal F})=0 \}$. This generalizes the well-studied parameter $La(n,P)=La(n,P,P_1)$ where $P_1$ is the one element poset. The quantity $La(n,P)$ has been determined (precisely or asymptotically) for many posets $P$, and in all known cases an asymptotically best construction can be obtained by taking as many middle levels as possible without creating a copy of $P$. In this paper we consider the first instances of the problem of determining $La(n,P,Q)$. We find its value when $P$ and $Q$ are small posets, like chains, forks, the $N$ poset and diamonds. Already these special cases show that the extremal families are completely different from those in the original $P$-free cases: sometimes not middle or consecutive levels maximize $La(n,P,Q)$ and sometimes no asymptotically extremal family is the union of levels. Finally, we determine the maximum number of copies of complete multi-level posets in $k$-Sperner families. The main tools for this are the profile polytope method and two extremal set system problems that are of independent interest: we maximize the number of $r$-tuples $A_1,A_2,\dots, A_r \in {\cal A}$ over all antichains ${\cal A}\subseteq 2^{[n]}$ such that (i) $\cap_{i=1}^rA_i=\emptyset$, (ii) $\cap_{i=1}^rA_i=\emptyset$ and $\cup_{i=1}^rA_i=[n]$.