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| suffices. Using a form of greedy algorithm we strengthen this to $$|G| > (2 + \sqrt 3 )|Y|^3 $$ . Some related results on finite and infinite groups are included. [ABSTRACT FROM AUTHOR]