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A very robust Ramsey theorem for matchings

Peter Keevash, Peleg Michaeli

Abstract

Our main result is a robust generalisation of the Cockayne-Lorimer theorem on the multicolour Ramsey number of matchings. It is moreover a generalisation of the transference generalisation of Cockayne-Lorimer, which (informally) says that the random graph $G \sim G(n,p)$ with $np \to \infty$ has, with high probability, essentially the same Ramsey matching properties as the complete graph $K_n$. We show, somewhat surprisingly, that the same is true under the rather weak robustness assumption that $G$ is an $s$-connector (i.e. $\overline{G}$ is $K_{s,s}$-free) with $s=o(n)$. Moreover, we show that such $G$ has only an additive $O(s)$ loss with respect to $K_n$ for monochromatic matchings, which is essentially sharp. Our proof adapts a compression algorithm based on Gallai-Edmonds decompositions that we developed previously for generalised Ramsey-Turán problems.

A very robust Ramsey theorem for matchings

Abstract

Our main result is a robust generalisation of the Cockayne-Lorimer theorem on the multicolour Ramsey number of matchings. It is moreover a generalisation of the transference generalisation of Cockayne-Lorimer, which (informally) says that the random graph with has, with high probability, essentially the same Ramsey matching properties as the complete graph . We show, somewhat surprisingly, that the same is true under the rather weak robustness assumption that is an -connector (i.e. is -free) with . Moreover, we show that such has only an additive loss with respect to for monochromatic matchings, which is essentially sharp. Our proof adapts a compression algorithm based on Gallai-Edmonds decompositions that we developed previously for generalised Ramsey-Turán problems.
Paper Structure (14 sections, 20 theorems, 14 equations, 4 algorithms)

This paper contains 14 sections, 20 theorems, 14 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Paper organisation.
  4. Single colour compressions
  5. Main algorithm
  6. Decycling
  7. A dominant component
  8. Distilling
  9. Proof of the main theorem
  10. Discussion
  11. Sharpness
  12. Transference
  13. Beating the pigeonhole bound
  14. Hypergraphs

Key Result

theorem 1

Let $q,s\ge 1$ be integers. For all $\mathbf{t}\in\mathbb{N}_+^q$ and $n\ge\|\mathbf{t}\|_\infty+\Lambda_\mathbf{t}+1+7(q+1)(s-1)$, if $G$ is an $n$-vertex $s$-connector then $G\to \mathbf{t} K_2$.

Theorems & Definitions (37)

  • theorem 1
  • proposition 2
  • theorem 3: Gallai--Edmonds Structure Theorem
  • lemma 4: Stability
  • lemma 5: $CD$-edges
  • proof
  • lemma 6: $CD$-saturation
  • proof
  • lemma 7: $C$-isolation
  • proof
  • ...and 27 more