The study of Ramsey properties of the binomial random graph G_{n,p} was initiated in the 80s by Frankl & Rödl and Łuczak, Ruciński & Voigt. In this area we are often interested in establishing the threshold function f(n) that governs G_{n,p} having a particular Ramsey-like property P or not, i.e. if p is sufficiently larger than f(n) then G_{n,p} asymptotically almost surely (a.a.s) has P, and if p is sufficiently smaller than f(n) then G_{n,p} a.a.s. does not have P.

In this talk, we will explore how combinatorial methods have been used to establish these thresholds for diagonal, off-diagonal and anti-Ramsey properties, including in recent joint work together with Natalie Behague, Candy Bowtell, Robert Hancock, Shoham Letzter and Natasha Morrison.