Intersecting families, signed sets, and injection

TL;DR

This paper proves a classical bound on the size of intersecting families of signed sets using an injection method, extending known combinatorial results with a new proof technique.

Contribution

It provides an injective proof of the maximum size of intersecting families of signed sets for certain parameters, offering a novel approach to a well-known combinatorial problem.

Findings

Established an injective proof for the maximum size bound

Extended known results to signed set families with new methodology

Identified open cases for further research

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 $k \leq n$.