Partitioning the edges of a complete graph into smaller complete graphs
After some search, I have found out that a solution was already published in 1948. De Bruijn and Erdos proved that it is not possible to partition the edges of $K_n$ into fewer than $n$ smaller cliques. I do not have a direct link to their paper, but it is referred to in the 1988 paper by Erdos, Faudree and Ordman, "Clique Partitions and Clique Coverings".