An improved algebraic construction for Ramsey numbers

We provide an explicit algebraic construction showing that, uniformly for integers $3\le s\le t$, as $t\to \mathrm{\infty }$, $$R(s,t)\ge t^{(1-o(1))\log s\mathrm{/}\log (\log s+1)}\mathrm{.}$$ For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudlák. In particular, $R(33,t)\ge t^{2.1-o(1)}$, to our knowledge, the first explicit construction showing $R(s,t)\ge t^c$ for some fixed $s$ and some $c>2$. In the diagonal case, it improves the leading constant in the exponent of the classical Frankl–Wilson bound from $1\mathrm{/}4$ to $1$, while being almost as simple to describe.