Abstract

We prove a new lower bound on the Ramsey number r(\ell, C\ell) for any constant C> 1 and sufficiently large \ell, showing that there exists \varepsilon(C) > 0 such that r(\ell, C\ell) ≥ \left(p_C^{-1/2} + \varepsilon(C)\right)^\ell, where p_C denotes the unique solution in (0, 1/2) satisfying C = \log p_C / \log (1 - p_C). This provides the first exponential improvement over the classical lower bound by Erdos since 1947. Joint work with Wujie Shen and Shengjie Xie.