Off-Diagonal Ramsey Multiplicity

06/30/2023

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by   Elena Moss, et al.

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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.