  In 1994 S. McGuinness showed that any greedy clique decompo- sition of an
n-vertex graph has at most $\lfloor n^2/4 \rfloor$ cliques (The greedy clique
decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique
decomposition means a clique partition of the edge set and a greedy clique
decomposition of a graph is obtained by removing maximal cliques from a graph
one by one until the graph is empty. This result solved a conjecture by P.
Winkler. A multifamily set rep- resentation of a simple graph G is a family of
sets, not necessarily distinct, each member of which represents a vertex in G,
and the in- tersection of two sets is non-empty if and only if two
corresponding vertices in G are adjacent. It is well known that for a graph G,
there is a one-to-one correspondence between multifamily set representations
and clique coverings of the edge set. Further for a graph one may have a
one-to-one correspondence between particular multifamily set rep- resentations
with intersection size at most one and clique partitions of the edge set. In
this paper, we study for an n-vertex graph the variant of the set
representations using a family of distinct sets, including the greedy way to
get the corresponding clique partition of the edge set of the graph. Similarly,
in this case, we obtain a result that any greedy clique decomposition of an
n-vertex graph has at most $\lfloor n^2/4 \rfloor$ cliques.
