Off-Diagonal Ramsey Multiplicity
For researchers in extremal combinatorics, this extends a classic problem to a new setting, providing foundational results and solutions for specific cases.
The paper introduces an off-diagonal generalization of the Ramsey multiplicity problem, minimizing a weighted sum of densities of red copies of one graph and blue copies of another, and solves the problem for several graph pairs.
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.