On The Conditional Hardness Of Coloring A 4-Colorable Graph With Super-Constant Number Of Colors

For 3 <= q < Q we consider the APPROXCOLORING(q, Q) problem of deciding whether chi(G) <= q or chi(G) >= Q for a given graph G. Hardness of this problem was shown in [7] for q = 3,4 and arbitrary large constant Q under variants of the Unique Games Conjecture [10].We extend this result to values of Q that depend on the size of a given graph. The extension depends on the parameters of the conjectures we consider. Following the approach of [7], we find that a careful calculation of the parameters gives hardness of coloring a 4-colorable graph with lg(c)(lg(n)) colors for some constant c > 0. By improving the analysis of the reduction we show that under related conjectures it is hard to color a 4-colorable graph with lg(c)(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 whose each coordinate has o(1) influence, the Majority function has the largest noise stability. We adapt the theorem for our applications to get a better dependency between the parameters required for the reduction.