A maximum [independent set](http://en.wikipedia.org/wiki/Independent_set_%28graph_theory%29) is a largest independent set for a given graph $G$ and its size is denoted $\alpha(G)$. And the [Lovász number](http://en.wikipedia.org/wiki/Lov%C3%A1sz_number) of $G$ is denoted $\vartheta(G)$. $\vartheta(G)\geq \alpha(G)$ by definition, then the question is what will the 
$$\max\limits\_G \frac{\vartheta(G)}{\alpha(G)}$$
be? Any known results about this topic will be welcome! One can give examples to show how big could the ratio $\vartheta(G)/\alpha(G)$ be or counter-examples to show the ratio will go to infinity. Any help or suggestions will be appreciated!


P.S. here is a related question:[Cliques, Paley graphs and quadratic residues](https://mathoverflow.net/questions/48591/cliques-paley-graphs-and-quadratic-residues).



What is the maximum of the ratio $\vartheta(G)/\alpha(G)$?
It is infinite, in fact much stronger versions are also true, see e.g., Theorem 1 here:
<http://arxiv.org/abs/cs/0608021>
(Shannon capacity is between $\alpha$ and $\vartheta$.)


