The Ramsey function $R(k,x)$ is defined as the minimal integer $n$ such that any graph on $n$ vertices contains either a clique of size $k$ or an independent set of size $x$. [Miklós Ajtai, János Komlós and Endre Szemerédi(1980)](https://linkinghub.elsevier.com/retrieve/pii/0097316580900308) shows that $R(3,x)≤cx^2/\ln x$ and further that, for each $k$, $R(k,x)\leq c\_kx^{k−1}/(\ln x)^{k−2}$, here $c\_k>0$ is a constant. It is well known that $c'x^2/\ln x \leq R(3,x)\leq cx^2/\ln x$ for some $c>c'>0$.


I want to know for $k\ge 4$, is there any better upper bounds for the Ramsey function $R(k,x)$? I care more about the order of the upper bound instead of the value of $c\_k$.



Best known upper bound for the Ramsey function $R(k,x)$
I don't have the reputation to comment, but I don't think there has been any improvement on the result you mentioned. See pg 5 of this survey by Conlon, Fox and Sudakov from 2015 <https://arxiv.org/pdf/1501.02474.pdf>


