Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cross-intersecting sub-families of hereditary families   Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said
to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$
with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in
$\mathcal{A}_j$. For a finite set $X$, let $2^X$ denote the \emph{power set of
$X$} (the family of all subsets of $X$). A family $\mathcal{H}$ is said to be
\emph{hereditary} if all subsets of any set in $\mathcal{H}$ are in
$\mathcal{H}$; so $\mathcal{H}$ is hereditary if and only if it is a union of
power sets. We conjecture that for any non-empty hereditary sub-family
$\mathcal{H} \neq \{\emptyset\}$ of $2^X$ and any $k \geq |X|+1$, both the sum
and product of sizes of $k$ cross-intersecting sub-families $\mathcal{A}_1,
\mathcal{A}_2, ..., \mathcal{A}_k$ (not necessarily distinct or non-empty) of
$\mathcal{H}$ are maxima if $\mathcal{A}_1 = \mathcal{A}_2 = ... =
\mathcal{A}_k = \mathcal{S}$ for some largest \emph{star $\mathcal{S}$ of
$\mathcal{H}$} (a sub-family of $\mathcal{H}$ whose sets have a common
element). We prove this for the case when $\mathcal{H}$ is \emph{compressed
with respect to an element $x$ of $X$}, and for this purpose we establish new
properties of the usual \emph{compression operation}. For the product, we
actually conjecture that the configuration $\mathcal{A}_1 = \mathcal{A}_2 = ...
= \mathcal{A}_k = \mathcal{S}$ is optimal for any hereditary $\mathcal{H}$ and
any $k \geq 2$, and we prove this for a special case too.

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The categories of this paper are: hep-ph
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th