I was asked about an elementary (math olympiad) problem regarding members and committees, which can be formulated as the following graph-theoretic problem.
Given a bipartite graph $(V_1\cup V_2, E)$, where the degree of each vertex in $V_1$ is no less than 1 and the degree of each vertex in $V_2$ is no less than 2. We assume that for any two vertices $u,v\in V_2$, there is at least one vertex in $V_1$ which is connected to both $u$ and $v$.
Show that the vertices of $V_1$ can be colored in three colors R, G, B, such that each vertex in $V_2$ is connected with at least two colors.
Choose the vertex $v \in V_2$ with minimal degree, such that no vertex in $V_2$ has adjacencies completely contained within $v$. Then, let $A = \{f : f \in V_1, f\text{ is adjacent to }v\}$, which is the set of all adjacent vertices to $v$ in $V_1$, and choose $g \in A$ to be colored red, while all other vertices in $A$ are colored blue. Then, color every other vertex in $V_1$ green. Since no vertex is contained completely within $v$, they should all have at least $1$ green vertex, and since every vertex in $V_2$ shares an adjacency with $v$, they also have one other color. Finally, $v$ has two colors since $g$ is red and the others are blue.