On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs

TL;DR

This paper advances bounds on multicolour hypergraph Ramsey numbers and explores hyperedge colouring constraints related to hypergraph chromatic number, providing new theoretical insights and bounds.

Contribution

It improves lower bounds on multicolour hypergraph Ramsey numbers and establishes bounds on the number of colours needed to colour hyperedges without large monochromatic hyperedges.

Findings

Improved lower bounds on multicolour hypergraph Ramsey numbers.

Derived bounds on colourings avoiding large monochromatic hyperedges.

Established a relationship between the number of colours and hypergraph chromatic number.

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 $\chi(H)$. In particular we show that this number is $O(\log^{(r-1)} (r \chi(H)) + r)$.