The r-colour size Ramsey number of the path P_k is the smallest m such that some m-edge graph G has a monochromatic P_k in every r-colouring of its edges. The linearity in k has been established by Beck in 1983, but finding the optimal r-dependence has been open since. The lower bound of cr^2 k by Krivelevich was suggested to be optimal, while the best upper bound of Dudek and Pralat is a factor of log r away. In this talk we introduce a new colouring technique that, somewhat surprisingly, yields a log r improvement in the lower bound, solving the problem up to constants. Joint work with Anqi Li and Julian Sahasrabudhe.
