  A graph $H$ is said to be common if the number of monochromatic labelled
copies of $H$ in a red/blue edge colouring of a large complete graph is
asymptotically minimized by a random colouring with an equal proportion of each
colour. We extend this notion to an asymmetric setting. That is, we define a
pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear
combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically
minimized by a random colouring in which each edge is coloured red with
probability $p$ and blue with probability $1-p$. We extend many of the results
on common graphs to this asymmetric setting. In addition, we obtain several
novel results for common pairs of graphs with no natural analogue in the
symmetric setting. We also obtain new examples of common graphs in the
classical sense and propose several open problems.
