An entropy proof of the commonality of apices of trees

A graph $H$ is called common if the number of monochromatic copies of $H$ in a $2$-edge-colouring of a large complete graph is asymptotically minimized by the random colouring with an equal proportion of each colour. Grzesik, Lee, Lidický and Volec conjectured that every $k$-apex of any connected Sidorenko graph is common. Recently, Kr{á}l’, Krnc and Lamaison proved that the $k$-apex of any tree is common for $k\le 5$. In this paper, we extend their result by showing that for every positive integer $k$, the $k$-apex of any tree or any forest with average degree at least one is common. Our proof involves an application of the entropy method.