Embedding graphs in Cayley graphs

If Y is a finite graph then it is known that every sufficiently large group G has a Cayley graph containing an induced subgraph isomorphic to Y. This raises the question as to what is "sufficiently large". Babai and Sós have used a probabilistic argument to show that |G| > 9.5 |Y|3 suffices. Using a form of greedy algorithm we strengthen this to {Mathematical expression}. Some related results on finite and infinite groups are included.