The Ramsey multiplicity problem asks for the minimum asymptotic density of monochromatic labelled copies of a graph $H$ in a red/blue colouring of the edges of $K_n$. We introduce an off-diagonal generalization in which the goal is to minimize a certain weighted sum of the densities of red copies of one graph and blue copies of another. We build up various properties of this new notion, including a useful "dual formulation," and use these results to solve the problem for several pairs of graphs.

翻译：拉姆齐重数问题关注的是，在$K_n$边的红/蓝着色中，单色标号图$H$的最小渐近密度。我们引入了一种非对角推广，其目标是使某个加权和最小化，该加权和由一种图的红色副本密度与另一种图的蓝色副本密度组成。我们建立了这一新概念的多种性质，包括一个有用的“对偶公式”，并利用这些结果解决了若干对图的问题。