Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\cup_{i=1}^rA_i=[n]$.

26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas