Consider the (simple, undirected) complete graph $K_n$. Is it possible to partition its edges into less than $n$ cliques, each having less than $n$ vertices?
Note: It is easy to see that $n$ cliques of size $<n$ always suffice. Take a $K_{n-1}$ as one clique, and take a $K_2$ for each of the $n-1$ edges that are not contained in the $K_{n-1}$. This gives an edge partition into $n$ cliques, each with size $<n$. The question is whether there is such a partition into less than $n$ cliques of size $<n$.