A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erd\H{o}s, conjectured that every graph is common. The conjectures by Erd\H{o}s and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s.
Despite its importance, the full classification of common graphs is still a wide open problem and has not seen much progress since the early 1990s. In this lecture, I will present some old and new techniques to prove whether a graph is common or not.
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