Intersecting family in a product alphabet ($\max |\mathcal{A}|=k^{n-1}$)

A family of words in $[k{]}^n$ is intersecting if any two words agree in some coordinate. For positive $n,k$, its maximum size is $k^{n-1}$: simultaneous cyclic shifts partition the words into classes in which distinct words never agree, and fixing one coordinate attains the resulting bound.