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Jul 24, 2013 at 11:26 comment added Ben Barber basil, Ramsey's theorem will also give you reasonable bounds on the clique number of $G$. If the clique numbers of $G'$ and $G''$ are $r$ and $s$, then the clique number of $G$ must be less than $R(r+1, s+1) \leq \binom {r+s} r$.
Jul 23, 2013 at 21:22 comment added Tarik Aougab thank you for your response- it seems Ramsey's theorem works quite well to answer the finiteness question.
Jul 23, 2013 at 21:19 vote accept Tarik Aougab
Jul 23, 2013 at 21:19
Jul 23, 2013 at 19:19 history answered Seva CC BY-SA 3.0