Thanks Joseph -- this is exactly what I was hoping to find out.

I don't know any general techniques to estimate the theta-function. However there are other, easier to compute, spectral lower bounds on chromatic number. In particular, see Theorem 5.2 of these notes: tinyurl.com/297vtgk

The De Bruijn–Erdős Theorem says that the chromatic number of an infinite graph (if it exists) is the maximum chromatic number of its finite subgraphs. Shelah and Soifer gave examples to suggest that the chromatic number of the plane might depend on set theoretic axioms. Here is an example of a coloring problem in the plane (with countably many colors), where the answer is equivalent to the Continuum Hypothesis: mathoverflow.net/questions/273/…

What is the easiest example of non-isomorphic finite groups $F$ and $G$ such that $F_g = F_h$?