Question 11
Let $k$ and $m$ be integers greater than $1$. Consider $k$ pairwise disjoint sets $S_1, S_2, \dots, S_k$; each of these sets has exactly $m+1$ elements, one of which is red and the other $m$ are all blue. Let $\mathcal{F}$ be the family of all subsets $F$ of $S_1 \cup S_2 \cup \dots \cup S_k$ such that, for every $i$, the intersection $F \cap S_i$ is monochromatic; the empty set is monochromatic. Determine the largest possible cardinality of a subfamily $\mathcal{G} \subseteq \mathcal{F}$, no two sets of which are disjoint.