Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and produce for this graph a certificate for the SDP which corresponds to having found a clique of size $n^\alpha$ in $G_1$ for some $\alpha \in (0,1]$. Now using this certificate (or any other means!) is it possible to construct another graph $G_2$  of size $N$ (might be necessary that $N>n$) and a certificate for it for this same SDP which would correspond to having found a $N^{ 1-\epsilon }$ size clique in $G_2$ for some $1 - \epsilon > \alpha$? (its okay even if $G_2$ is shown to exist in some probabilistic sense) Are there examples of doing such a blow-up of the objective value for a Max-Clique SDP?Not sure if its immediately related but putting this out here : Is there any obvious reason why this very famous paper http://ttic.uchicago.edu/~madhurt/Papers/reductions.pdf does not deal with Max-Clique? Isn't it true that the complement of the Max-Independent-Set integrality gap instance does not give the integrality gap instance of Max-Clique? 
About increasing the objective values of certificates for Max-Clique SDP
cc.complexity theory;np hardness;complexity classes;approximation hardness;sum of squares