On the conditional hardness of coloring a 4-colorable graph with superconstant number of colors

For 3 ≤ q < Q we consider the ApproxColoring(q,Q) problem of deciding for a given graph G whether χ(G) ≤ q or χ(G) ≥ Q. It was show in [DMR09] that the problem ApproxColoring(q,Q) is NP-hard for q = 3, 4 and arbitrary large constant Q under variants of the Unique Games Conjecture. In this paper we give a tighter analysis of the reduction of [DMR09] from Unique Games to the ApproxColoring problem. We find that (under appropriate conjecture) a careful calculation of the parameters in [DMR09] implies hardness of coloring a 4-colorable graph with logc(log(n)) colors for some constant c> 0. By improving the analysis of the reduction we show hardness of coloring a 4-colorable graph with logc(n) colors for some constant c> 0. The main technical contribution of the paper is a variant of the Majority is Stablest Theorem, which says that among all balanced functions in which every coordinate has o(1) influence, the Majority function has the largest noise stability. We adapt the the-orem for our applications to get a better dependency between the parameters required for the reduction