Size Ramsey Numbers: A tale of three families

The size Ramsey number of a graph H is defined as the minimum number of edges in a graph G such that there is a monochromatic copy of H in every two-coloring of E(G). The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs, and starburst graphs. In this talk, we completely resolve the latter two questions and make substantial progress on the first by determining up to a constant factor the size Ramsey number of complete bipartite graphs with parts of size s and t for all t = Ω(s logs). The proofs involve a variety of random graph models and probabilistic methods. 

Based on joint work with David Conlon and Yuval Wigderson.