Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity
Joseph Hyde, Jae-baek Lee, Jonathan A. Noel

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 is the limit as tends to infinity of the minimum density of monochromatic labeled copies of in a -edge colouring of . 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
