Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity

The Ramsey multiplicity constant of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labelled copies of $H$ in a colouring of the edges of $K_n$ with two colours. Fox and Wigderson recently identified a large family of graphs whose Ramsey multiplicity constants are attained by sequences of "Tur\'an colourings;" i.e. colourings in which one of the colour classes forms the edge set of a balanced complete multipartite graph. The graphs in their family come from taking a connected non-3-colourable graph with a critical edge and adding many pendant edges. We extend their result to an off-diagonal variant of the Ramsey multiplicity constant which involves minimizing a weighted sum of red copies of one graph and blue copies of another. We also apply the flag algebra method to investigate the minimum number of pendant edges required for Tur\'an colourings to become optimal when the underlying graphs are small cliques.