Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 130 / 1 / ii

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, provingIf $\mathcal{F}$ is empty there is no permitted active set at all, so it is also not an adequate family of active coordinate sets.