  We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$
or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional
hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our
result is based on Khot's 2-to-1 conjecture [Khot'02]. For $q=3$, we base our
hardness result on a certain `fish shaped' variant of his conjecture.
  We also prove that the problem almost coloring is hard for any constant
$\eps>0$, assuming Khot's Unique Games conjecture. This is the problem of
deciding for a given graph, between the case where one can 3-color all but a
$\eps$ fraction of the vertices without monochromatic edges, and the case where
the graph contains no independent set of relative size at least $\eps$.
  Our result is based on bounding various generalized noise-stability
quantities using the invariance principle of Mossel et al [MOO'05].
