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A canonical Ramsey theorem with list constraints in random (hyper-)graphs

José D. Alvarado, Yoshiharu Kohayakawa, Patrick Morris, Guilherme O. Mota

Abstract

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise to copies of $H$ that exhibit certain colour patterns. We are interested in sparse random versions of this result and the threshold at which the random $k$-graph ${\mathbf{G}}^{(k)}(n,p)$ inherits the canonical Ramsey properties of $K^{(k)}_n$. Our main result here pins down this threshold when we focus on colourings that are constrained by some prefixed lists. This result is applied in an accompanying work of the authors on the threshold for the canonical Ramsey property (with no list constraints) in the case that $H$ is a (2-uniform) even cycle.

A canonical Ramsey theorem with list constraints in random (hyper-)graphs

Abstract

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given -uniform hypergraph (or -graph) , if is sufficiently large then any colouring of the edges of the complete -graph gives rise to copies of that exhibit certain colour patterns. We are interested in sparse random versions of this result and the threshold at which the random -graph inherits the canonical Ramsey properties of . Our main result here pins down this threshold when we focus on colourings that are constrained by some prefixed lists. This result is applied in an accompanying work of the authors on the threshold for the canonical Ramsey property (with no list constraints) in the case that is a (2-uniform) even cycle.
Paper Structure (17 sections, 11 theorems, 13 equations)

This paper contains 17 sections, 11 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Sparse Ramsey theory and random graphs
  3. Canonical Ramsey properties of random graphs
  4. An application for even cycles
  5. Hypergraphs
  6. Alternative characterisations of the canonical Ramsey property
  7. Related work
  8. Organisation
  9. Preliminaries
  10. The method of containers
  11. Locally dense graphs
  12. Outline of the proof
  13. Proof
  14. A hypergraph encoding canonical copies
  15. Supersaturation for canonical copies
  16. ...and 2 more sections

Key Result

Theorem 1.4

Let $r\geq 2$ be an integer and $H$ be a nonempty graph which is not a star forest. Then $n^{-1/m_2(H)}$ is the threshold for the property that ${{\mathbf G}(n,p) \xrightarrow[{\raisebox{.5mm}{$\rm$}}]{\raisebox{0.0mm}{$r$}} H}$.

Theorems & Definitions (26)

  • Definition 1.1: Lexicographic colouring
  • Definition 1.2: The canonical Ramsey property
  • Definition 1.3
  • Theorem 1.4: Rödl--Ruciński RR93RR94RR95
  • Definition 1.5: Colourings with list constraints
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8: Projection maps
  • Definition 1.9: Canonical copies
  • Theorem 1.10
  • ...and 16 more