Alon's conjecture on Ramsey Cayley graphs

Let $G$ be a finite group. A Cayley graph of $G$ is formed from a symmetric subset $S\subset G$ by joining $x,y\in G$ when $x^{-1}y\in S$. Such a graph is $C$-Ramsey if it has no clique or independent set of size larger than $C\log_2|G|$. Alon's conjecture. There exists some absolute constant $C>0$ for which every finite group has a $C$-Ramsey Cayley graph. The conjecture is the central motivation of the paper; the authors confirm it for groups of almost all orders, but the general case remains open.