This question tests understanding of set intersection operations. The key insight is that pairwise intersections can be non-empty while the triple intersection remains empty. This occurs when each pair shares a unique element not present in the third set.

Using $A = \{1, 2\}$, $B = \{2, 3\}$, $C = \{1, 3\}$ creates a situation where element $2$ is shared by $A$ and $B$ only, element $3$ is shared by $B$ and $C$ only, and element $1$ is shared by $A$ and $C$ only. No element belongs to all three sets, making $A \cap B \cap C = \phi$.

This satisfies all conditions: each pairwise intersection contains exactly one element, while the triple intersection is empty.