  For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs
exist, then the Ramsey number
  $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves
the best lower bounds for $r(C_{\ell},t)$ obtained by Bohman and Keevash from
the random $C_{\ell}$-free process by polylogarithmic factors for all odd $\ell
\geq 5$ and $\ell \in \{6,10\}$. For $\ell = 4$ it matches their lower bound
from the $C_4$-free process.
  We also prove, via a different approach, that $r(C_5, t)> (1+o(1))t^{11/8}$
and $r(C_7, t)> (1+o(1))t^{11/9}$. These improve the exponent of $t$ in the
previous best results and appear to be the first examples of graphs $F$ with
cycles for which such an improvement of the exponent for $r(F, t)$ is shown
over the bounds given by the random $F$-free process and random graphs.
