Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intersections and Distinct Intersections in Cross-intersecting Families   Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of
$k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$,
$\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections
$F\cap G$ with $F\in \mathcal{F},G\in \mathcal{G}$, and all distinct
intersections $F\cap G$ with $F\neq G, F\in \mathcal{F},G\in \mathcal{G}$,
respectively. For a fixed $T\subset \{1,2,\ldots,n\}$, let $\mathcal{S}_T$ be
the family of all $k$-subsets of $\{1,2,\ldots,n\}$ containing $T$. In the
present paper, we show that $|\mathcal{F}\wedge \mathcal{G}|$ is maximized when
$\mathcal{F}=\mathcal{G}=\mathcal{S}_{\{1\}}$ for $n\geq 2k^2+8k$, while
surprisingly $|\mathcal{I}(\mathcal{F}, \mathcal{G})|$ is maximized when
$\mathcal{F}=\mathcal{S}_{\{1,2\}}\cup \mathcal{S}_{\{3,4\}}\cup
\mathcal{S}_{\{1,4,5\}}\cup \mathcal{S}_{\{2,3,6\}}$ and
$\mathcal{G}=\mathcal{S}_{\{1,3\}}\cup \mathcal{S}_{\{2,4\}}\cup
\mathcal{S}_{\{1,4,6\}}\cup \mathcal{S}_{\{2,3,5\}}$ for $n\geq 100k^2$. The
maximum number of distinct intersections in a $t$-intersecting family is
determined for $n\geq 3(t+2)^3k^2$ as well.

15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con