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Published online by Cambridge University Press: 30 June 2025
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 minimised by a random colouring in which each edge is equally likely to be red or blue. We extend this notion to an off-diagonal 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 minimised by a random colouring in which each edge is coloured red with probability
$p$ and blue with probability
$1-p$. Our results include off-diagonal extensions of several standard theorems on common graphs and novel results for common pairs of graphs with no natural analogue in the classical setting.