  We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$,
the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some
part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$
such that $A\cap B$, $A\cap C$, and $B\cap C$ have distinct representatives.
This is sharp, since by placing two complementary pairs of sets into each
partition class, we have a partition into $2^{n-2}$ triangle-free parts. We
also address a more general Ramsey-type problem: for a given graph $G$, find
(estimate) $f(n,G)$, the smallest number of colors needed for a coloring of
$\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph.
We give an upper bound for $f(n,G)$ for any connected graph $G$ which is
asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or
star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.
