The Ramsey number r(s,t) denotes the minimum N such that in any red-blue coloring of the edges of the complete graph on N vertices, there exists a red complete graph on s vertices or a blue complete graph on t vertices. While the study of these quantities goes back almost one hundred years, to early papers of Ramsey and Erdos and Szekeres, the long-standing conjecture of Erdos that r(s,t) has order of magnitude at most a polynomial in t of degree s - 1 remains open in general. It took roughly sixty years before the order of magnitude of r(3,t) was determined by Jeong Han Kim. In this talk, we discuss a variety of new techniques which lead to the solution of the conjecture for r(4,t). We also come close to determining related quantities known as Erdos-Rogers functions, as well as determining the current best bounds for other graph Ramsey numbers. Joint work in part with Sam Mattheus and Dhruv Mubayi.