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On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs

Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, Yelena Yuditsky

Abstract

For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous results by \textcite{AGLM14} and \textcite{EHMR84} we show that \[ \text{if } r \geq 3 \text{ and } n \nrightarrow (s)_k^r \text{ then } 2^n \nrightarrow (s+1)_{k+3}^{r+1}. \] This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of colouring all $r$-subsets of $V$ such that no hyperedge of size $\geq r+1$ is monochromatic. We provide upper and lower bounds on the number of colours necessary in terms of the chromatic number $χ(H)$. In particular we show that this number is $O(\log^{(r-1)} (r χ(H)) + r)$.

On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs

Abstract

For and , write if every hyperedge colouring with colours of the complete -uniform hypergraph on vertices has a monochromatic subset of size . Improving upon previous results by \textcite{AGLM14} and \textcite{EHMR84} we show that This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph , we consider the Ramsey-like problem of colouring all -subsets of such that no hyperedge of size is monochromatic. We provide upper and lower bounds on the number of colours necessary in terms of the chromatic number . In particular we show that this number is .

Paper Structure

This paper contains 17 sections, 10 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations.
  3. Subset colouring in hypergraphs.
  4. Previous results
  5. Organization of the paper
  6. A stepping-up lemma
  7. Preliminary results and definitions
  8. Splitting indices.
  9. Caterpillars.
  10. Description and validity of the colouring.
  11. Description.
  12. Validity.
  13. Lower bounds on multicolour hypergraph Ramsey numbers
  14. Subset-colouring of hypergraphs
  15. Validity of the colouring.
  16. ...and 2 more sections

Key Result

Theorem 1.2

Fix integers $k\geq 2$ and $n>r\geq 3$ and $k$ integers $(s_i)_{i\in\llbracket k \rrbracket}$ all satisfying $n\geq s_i \geq r+1$. with the integer

Figures (1)

  • Figure 1: The tree structures of two caterpillars (left) and a non-caterpillar (right). The splitting indices are marked in red and circled.

Theorems & Definitions (26)

  • Definition 1.1: Rado's arrow notation
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • proof
  • proof
  • proof
  • proof
  • Theorem 3.2: ER52ER52, EHR65EHR65
  • Theorem 3.3: AGLM14
  • ...and 16 more