Graph products and chromatic numbers

The problem of computing the chromatic number of a graph is considered. No known approximation algorithm can guarantee a better than O(n^0.4) coloring on a three-chromatic graph with n vertices. Evidence is provided that it is inherently impossible to achieve a better than n^ε ratio in polynomial time by showing that `breaking the n^ε barrier´ will automatically lead to vastly better polynomial-time approximation algorithms that achieve ratios closer to log n