= Solution
Let $\mathcal F$ be an <intersecting family>. For a word $w\in[m]^n$, put $S(w)=\{i:w_i=1\}$. Give $w$ the red color if $S(w)$ contains some member of $\mathcal F$, and the blue color otherwise.
Consider any <combinatorial line> with active set $A\in\mathcal F$. At variable value $1$, $A\subseteq S(w)$, so that word is red. At variable value $2$, $S(w)\cap A=\varnothing$. No member of $\mathcal F$ can be contained in this $S(w)$, because every member meets $A$. Thus that word is blue. No such line is <monochromatic>, proving
$$
\boxed{\mathcal F\text{ intersecting }\Longrightarrow\mathcal F\text{ not adequate}.}
$$
If $\mathcal F$ is empty there is no permitted active set at all, so it is also not an <adequate family of active coordinate sets>.
Back to article page
