RMM 2025 Problem 6

Let $k$ and $m$ be integers greater than 1. Consider $k$ pairwise disjoint sets $S_{1}, S_{2}, \ldots , S_{k}$ , each of which has exactly $m + 1$ elements: one red and $m$ blue. Let $\mathcal{F}$ be the family of all subsets $T$ of $S_{1} \cup S_{2} \cup \dots \cup S_{k}$ such that, for every $i$ , the intersection $T \cap S_{i}$ is monochromatic. Determine the largest possible number of sets in a subfamily $\mathcal{G} \subseteq \mathcal{F}$ such that no two sets in $\mathcal{G}$ are disjoint.

A set is monochromatic if all of its elements have the same colour. In particular, the empty set is monochromatic.

Each problem is worth 7 marks. Time allowed: $4\frac{1}{2}$ hours.