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Turán Colourings in Off-Diagonal Ramsey Multiplicity

Joseph Hyde, Jae-baek Lee, Jonathan A. Noel

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 ``Turán 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.

Turán Colourings in Off-Diagonal Ramsey Multiplicity

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 ``Turán 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.
Paper Structure (10 sections, 26 theorems, 178 equations)

This paper contains 10 sections, 26 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Formal Definitions
  3. Proof of Theorem \ref{['th:hairy']}: Rough Structure
  4. Analyzing Degrees
  5. Obtaining the Partition
  6. Proof of Theorem \ref{['th:hairy']}: Exact Structure
  7. Every Vertex Contributes the Same
  8. Refining the Partition
  9. Beating the Turán Colouring
  10. Conclusion

Key Result

Theorem 1.1

For any connected non-3-colourable graph $F$ that contains a critical edge, there exists $t_0=t_0(F)$ such that, for any $t\geq t_0$, every $t$-hairy $F$ is a bonbon.

Theorems & Definitions (55)

  • Theorem 1.1: Fox and Wigderson FoxWigderson23
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.2: Fox and Wigderson FoxWigderson23
  • Definition 2.3
  • Definition 2.4: Moss and Noel MossNoel23++
  • Remark 3.1
  • Definition 3.5
  • Lemma 3.7
  • Lemma 3.9
  • ...and 45 more