Is the problem "MIN-SET-PARTITION" an NP-hard problem?
Take $S_i$ to be the edges of a graph $G$. Then, a partition $F$ of size $k$ corresponds to an edge-coloring of $G$; the minimum such $k$ corresponds to the chromatic index of $G$. Edge-coloring is known to be NP-complete: "The NP-Completeness of Edge-Colouring".