Adequate family of active coordinate sets

For a fixed alphabet size $m>1$, a family $\mathcal{F}\subseteq \mathcal{P}([n])$ is adequate if every two-color finite coloring of $[m{]}^n$ has a monochromatic combinatorial line whose nonempty active coordinate set belongs to $\mathcal{F}$.

An intersecting family is never adequate. Color a word red when the coordinates carrying letter $1$ contain a member of $\mathcal{F}$, and blue otherwise. For a line with active set $A\in \mathcal{F}$, its letter-$1$ word is red and its letter-$2$ word is blue, because no member of an intersecting family can lie wholly outside $A$.