An exponential improvement for Ramsey lower bounds

We prove a new lower bound on the Ramsey number $r(\mathrm{\ell },C\mathrm{\ell })$ for any constant $C>1$ and sufficiently large $\mathrm{\ell }$, showing that there exists $\epsilon =\epsilon (C)>0$ such that $$r(\mathrm{\ell },C\mathrm{\ell })\ge {(p_C^{-1\mathrm{/}2}+\epsilon )}^{\mathrm{\ell }},$$ where $p_C\in (0,1\mathrm{/}2)$ is the unique solution to $C=\frac{\log p_C}{\log (1-p_C)}$. This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947.