Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intersecting families, signed sets, and injection   Let $k, r, n \geq 1$ be integers, and let $\S_{n, k, r}$ be the family of
$r$-signed $k$-sets on $[n] = \{1, \dots, n\}$ given by $$ \mathcal{S}_{n, k,
r} = \Big\{\{(x_1, a_1), \dots, (x_k, a_k)\}: \{x_1, \dots, x_k\} \in
\binom{[n]}{k}, a_1, \dots, a_k \in [r] \Big\}. $$ A family $\mathcal{A}
\subseteq \S_{n, k, r}$ is \emph{intersecting} if $A, B \in \mathcal{A}$
implies $A \cap B \not= \emptyset$. A well-known result (first stated by Meyer
and proved using different methods by Deza and Frankl, and Bollob\'as and
Leader) states that if $\mathcal{A} \subseteq \mathcal{S}_{n, k, r}$ is
intersecting, $r \geq 2$ and $1 \leq k \leq n$, then $$|\mathcal{A}| \leq
r^{k-1}\binom{n-1}{k - 1}.$$ We provide a proof of this result by injection (in
the same spirit as Frankl and F\"uredi's and Hurlbert and Kamat's injective
proofs of the Erd\H{o}s--Ko--Rado Theorem, and Frankl's and Hurlbert and
Kamat's injective proofs of the Hilton--Milner Theorem) whenever $r \geq 2$ and
$1 \leq k \leq n/2$, leaving open only some cases when $k \leq n$.

11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph