The semi-inducibility problem

Let $H$ be a $k$-edge-coloured graph and let $n$ be a positive integer. What is the maximum number of copies of $H$ in a $k$-edge-coloured complete graph on $n$ vertices? This paper studies the case $k=2$, which we call the semi-inducibility problem. This problem is a generalisation of the inducibility problem of Pippenger and Golumbic which is solved only for some small graphs and limited families of graphs. We prove sharp or almost sharp results for alternating walks, for alternating cycles of length divisible by 4, and for 4-cycles of every colour pattern. Liu, Mubayi and Reiher asked whether there is a graph $F$ for which the binomial random graph is an asymptotically extremal graph in the inducibility problem over all graphs of a given edge density. This was recently answered in a strong negative sense by Jain, Michelen and Wei. In contrast, we find a \emph{quantum} graph $Q$ with positive coefficients and an interval of edge densities for which the only extremal graphs are quasirandom.