Large non-trivial $t$-intersecting families for signed sets

TL;DR

This paper characterizes the structure and maximum size of large non-trivial t-intersecting families of signed sets, extending known results to more complex set families.

Contribution

It provides a complete description of large maximal non-trivial t-intersecting families of signed sets, generalizing Hilton-Milner results.

Findings

Identifies the structure of large maximal non-trivial t-intersecting families

Determines maximum size of such families for t ≥ 2

Extends Hilton-Milner-type results to signed sets

Abstract

For positive integers $n,r,k$ with $n\ge r$ and $k\ge2$, a set $\{(x_1,y_1),(x_2,y_2),\dots,(x_r,y_r)\}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,\dots,x_r$ are distinct elements of $[n]$ and $y_1\dots,y_r\in[k]$. We say a $t$-intersecting family consisting of $k$-signed $r$-sets on $[n]$ is trivial if each member of this family contains a fixed $k$-signed $t$-set. In this paper, we determine the structure of large maximal non-trivial $t$-intersecting families. In particular, we characterize the non-trivial $t$-intersecting families with maximum size for $t\ge2$, extending a Hilton-Milner-type result for signed sets given by Borg.