The size Ramsey number of a graph $H$ is the minimum number of edges in a graph $G$ with the property that no matter how we two-color the edges of $G$, we can find a monochromatic copy of $H$. This notion was introduced in 1978 by Erdős, Faudree, Rousseau, and Schelp, and despite more than four decades of work, there is a lot that is still unknown; notably, of the four questions that conclude the Erdős-Faudree-Rousseau-Schelp paper, only one had been resolved as of last year. In this talk, I'll discuss recent work in which we resolve $\approx 2.5$ of the three remaining questions, using a variety of new combinatorial and probabilistic constructions.
Based on joint work with David Conlon and Jacob Fox.