Rainbow Ramsey simple structures

TL;DR

This paper investigates rainbow Ramsey properties of ultrahomogeneous structures like the Rado graph, demonstrating their existence and implications for finite graph colorings and embeddings.

Contribution

It proves certain ultrahomogeneous structures are rainbow Ramsey and derives new finite graph coloring and embedding results using compactness.

Findings

Rado graph is rainbow Ramsey

Existence of graphs with specific coloring properties

Implications for finite graph embeddings and colorings

Abstract

A relational structure $\mathrm{R}$ is {\em rainbow Ramsey} if for every finite induced substructure $\mathrm{C}$ of $\mathrm{R}$ and every colouring of the copies of $\mathrm{C}$ with countably many colours, such that each colour is used at most $k$ times for a fixed $k$, there exists a copy $\mathrm{R}^\ast$ of $\mathrm{R}$ so that the copies of $\mathrm{C}$ in $\mathrm{R^\ast}$ use each colour at most once. We show that certain ultrahomogenous binary relational structures, for example the Rado graph, are rainbow Ramsey. Via compactness this then implies that for all finite graphs $\mathrm{B}$ and $\mathrm{C}$ and $k \in \omega$, there exists a graph $\mathrm{A}$ so that for every colouring of the copies of $\mathrm{C}$ in $\mathrm{A}$ such that each colour is used at most $k$ times, there exists a copy $\mathrm{B}^\ast$ of $\mathrm{B}$ in $\mathrm{A}$ so that the copies of $\mathrm{C}$ in $\mathrm{B^\ast}$ use each colour at most once.