Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp results concerning disjoint cross-intersecting families   For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its
$k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called
cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$,
$B\in\mathcal B$. Let $f(n,k)$ denote the maximum of $\min\{|\mathcal A|,
|\mathcal B|\}$ where the maximum is taken over all pairs of {\em disjoint},
cross-intersecting families $\mathcal A, \mathcal B\subset\binom{[n]}{k}$. Let
$c=\log_2e$. We prove that
$f(n,k)=\left\lfloor\frac12\binom{n-1}{k-1}\right\rfloor$ essentially iff
$n>ck^2$ (cf. Theorem~1.4 for the exact statement). Let $f^*(n,k)$ denote the
same maximum under the additional restriction that the intersection of all
members of both $\mathcal A$ and $\mathcal B$ are empty. For $k\ge5$ and $n\ge
k^3$ we show that
$f^*(n,k)=\left\lfloor\frac12\left(\binom{n-1}{k-1}-\binom{n-2k}{k-1}\right)\right\rfloor+1$
and the restriction on $n$ is essentially sharp (cf. Theorem~5.4).

20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol