Coloring 3-colorable graphs with $O(n^{4/23})$ colors via a Gaussian-cover recursion

Abstract: We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $$\smash{O(n^{4/23}) = O(n<sup>{0.17391\ldots})}$$ colors, improving on the recent bounds of $O(n<sup>{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $$O(n^{{(13-\sqrt{97})/18+ε})=O(n^{{0.17506\dots}} + ε})$$ colors for every fixed $\smash{ε&gt;0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $μ$, and variance $v$, we prove $v\leq r^{2-μ^2/(2\log} m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.