Intersecting families, signed sets, and injection

Abstract

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 $k \leq n$.