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

Abstract

For positive integers $n,r,k$ with $n\ge r$ and $k\ge 2$, 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 $t\ge 2$, extending a Hilton-Milner-type result for signed sets given by Borg.