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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 $<n$.

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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".

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    $\begingroup$ The 1948 paper is here. $\endgroup$ Commented May 25, 2015 at 3:49

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