A note on Ramsey numbers

Upper bounds are found for the Ramsey function. We prove $\text{R(3, x) <}\text{cx}{}^2\text{ln}\text{x}$ and, for each k ? 3, $\text{R(k, x) <}\text{c}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k\; ?\; 1}}\text{(}\text{ln}\text{x)}{}^{\mathrm{k\; ?\; 2}}$ asymptotically in x.