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 base 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 “⋉-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.