Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cross-Sperner families   A pair of families $(\cF,\cG)$ is said to be \emph{cross-Sperner} if there
exists no pair of sets $F \in \cF, G \in \cG$ with $F \subseteq G$ or $G
\subseteq F$. There are two ways to measure the size of the pair $(\cF,\cG)$:
with the sum $|\cF|+|\cG|$ or with the product $|\cF|\cdot |\cG|$. We show that
if $\cF, \cG \subseteq 2^{[n]}$, then $|\cF||\cG| \le 2^{2n-4}$ and
$|\cF|+|\cG|$ is maximal if $\cF$ or $\cG$ consists of exactly one set of size
$\lceil n/2 \rceil$ provided the size of the ground set $n$ is large enough and
both $\cF$ and $\cG$ are non-empty.

13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int