Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniformly cross intersecting families   Let $\mathcal{A}$ and $\matchcal{B}$ denote two families of subsets of an
$n$-element set. The pair $(\mathcal{A},\mathcal{B})$ is said to be
$\ell$-cross-intersecting iff $|A\cap B| = \ell$ for all $A\in\mathcal{A}$ and
$B\in\mathcal{B}$. Denote by $P_\ell(n)$ the maximum value of
$|\mathcal{A}||\mathcal{B}|$ over all such pairs. The best known upper bound on
$P_\ell(n)$ is $\Theta(2^n)$, by Frankl and R\"{o}dl. For a lower bound,
Ahlswede, Cai and Zhang showed, for all $n \geq 2\ell$, a simple construction
of an $\ell$-cross-intersecting pair $(\mathcal{A},\mathcal{B})$ with
$|\mathcal{A}||\mathcal{B}| =
\binom{2\ell}{\ell}2^{n-2\ell}=\Theta(2^n/\sqrt{\ell})$, and conjectured that
this is best possible. Consequently, Sgall asked whether or not $P_\ell(n)$
decreases with $\ell$.
  In this paper, we confirm the above conjecture of Ahlswede et al. for any
sufficiently large $\ell$, implying a positive answer to the above question of
Sgall as well. By analyzing the linear spaces of the characteristic vectors of
$\mathcal{A},\mathcal{B}$ over $\mathbb{R}$, we show that there exists some
$\ell_0>0$, such that $P_\ell(n) \leq \binom{2\ell}{\ell}2^{n-2\ell}$ for all
$\ell \geq \ell_0$. Furthermore, we determine the precise structure of all the
pairs of families which attain this maximum.

31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas