New upper bound for multicolor Ramsey numbers

Let $R_r(k)$ denote the diagonal $r$-color graph Ramsey number. We prove that there exist absolute constants $c,K>0$ such that $$R_r(k)\le \exp\!\left(-c\frac{k}{r^2\log^4(2r)}\right)r^{rk}$$ for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. The proof combines a positive-coefficient root filter of variable order with a retained-spine refinement of the multicolor book method.b