Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity

TL;DR

This paper extends the concept of Turán colourings to off-diagonal Ramsey multiplicity constants, identifying optimal colorings for minimizing weighted sums of monochromatic subgraphs in edge colorings.

Contribution

It generalizes previous results by Fox and Wigderson to off-diagonal cases involving weighted sums of different monochromatic subgraphs.

Findings

Identifies Turán colourings as optimal for off-diagonal Ramsey multiplicity.

Extends previous diagonal results to more general off-diagonal scenarios.

Provides new bounds and characterizations for weighted monochromatic subgraph densities.

Abstract

The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. 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. Each graph in their family comes 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.