AoPS - Problem

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

$n+1$