= 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 $A$.
Back to article page
