Rainbow Colourings of Chains

Abstract

We prove that if $m$ be a positive integer and $X$ is a totally ordered set, then there exists a function $\varphi :X\to \{1,\dots ,m\}$ such that, for every interval $I$ in $X$ and every positive integer $r\le |I|$, there exist elements $x_1<x_2<\cdots <x_r$ of $I$ such that $\varphi (x_{i+1})\equiv \varphi (x_i)+1(\mathrm{mod}m)$ for $i=1,\dots ,r-1$.