Elementary Proofs of Commonality of the House Graph and $K_4$ with One Edge Subdivided

Source abstract

A graph $H$ is called common if the number of monochromatic copies of $H$ in a $2$-edge-coloring of a large complete graph is asymptotically minimized by the random coloring with an equal proportion of each color. The house graph and $K_4$ with one edge subdivided once are known to be common by the flag algebra method. Grzesik, Lee, Lidický, and Volec asked for ``human-friendly'' proofs of the commonality of these two graphs. In this short note, we give flag-algebra-free proofs that both graphs are common.