Off-Diagonal Ramsey Multiplicity

TL;DR

This paper introduces an off-diagonal generalization of the Ramsey multiplicity problem, focusing on minimizing a weighted sum of monochromatic subgraph densities in edge-colored complete graphs, and provides solutions for specific graph pairs.

Contribution

It proposes a novel off-diagonal Ramsey multiplicity framework, develops its theoretical properties, including a dual formulation, and solves the problem for several graph pairs.

Findings

Developed a dual formulation of the off-diagonal problem

Solved the problem for multiple pairs of graphs

Established properties of the new off-diagonal notion

Abstract

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.