WLOG $S=\{1,2,\dots,n\}$ Lemma: if $B$ is a subset of $A$ and is not equal to $A$, then $A$ and $B$ are in different parts. Consider the $n+1$ subsets: $$\{\} \{1\} \{1,2\} \{1,2,3\} \dots \{1,2, \dots,n\}$$According to the lemma any two of these subsets must be in different parts therefore $m \geq n+1$ and for $m=n+1$ we have: $S_0=\{\{\}\}$ $S_i=$ the set of all $i$ element subsets of $S$ So the minimum of $m$ is $n+1$