Conditional Hardness for Approximate Coloring

We study the AprxColoring$(q,Q)$ problem: Given a graph G, decide whether χ(G)≤ q or χ(G)≥ Q. We present hardness results for this problem for any constants 3≤ q<Q. For q≥4, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775]. For $q=3$, we base our hardness result on a certain “-0.5em<-shaped” variant of his conjecture. Previously no hardness result was known for $q=3$ and Q≥6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz [“Noise stability of functions with low influences: Invariance and optimality,” Ann. of Math. (2), to appear] and should have wider applicability.