Bibliographic Details
| Title: |
Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity |
| Authors: |
Hyde, Joseph, Lee, Jae-baek, Noel, Jonathan A. |
| Publication Year: |
2023 |
| Collection: |
Mathematics |
| Subject Terms: |
Mathematics - Combinatorics, 05C35, 05D10 |
| More Details: |
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.
Comment: 33 pages. Note that the first arXiv version of this paper contained several Ramsey multiplicity results for pairs of small graphs proved via the flag algebra method. We have decided to remove those results to make the paper shorter and more focused. We are currently preparing a second manuscript in which we plan to include generalizations of some of the statements that we removed |
| Document Type: |
Working Paper |
| Access URL: |
http://arxiv.org/abs/2309.06959 |
| Accession Number: |
edsarx.2309.06959 |
| Database: |
arXiv |
