Decompose a complete graph into smaller cliques
If you want to decompose a complete graph into triangles then look for "steiner triple system" (STS). You will find some there: http://steinertriples.fr/

If you want to decompose a complete graph into cliques of size p, the generalization of STS that you want are "Balanced Incomplete Block Designs". If you want to know more about that, try to find the book from Douglas Stinson named "Combinatorial designs: construction and analysis".

Finally, if you want to build some BIBD, we've been sweating a lot over combinatorial designs code in Sage recently. It can build a (v,k,1)-BIBD whenever one exists for k&lt;=5, and as for larger values of k, well, it depends.

Either way, if you find a combinatorial design that it cannot build come tell us about it ! :-P

Nathann