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 $k \leq n$.
Categories: math.CO