On sequences covering all rainbow $k$-progressions

Abstract

Let $\text{ac}(n,k)$ denote the smallest positive integer with the property that there exists an $n$-colouring $f$ of $\{1,\dots ,\text{ac}(n,k)\}$ such that for every $k$-subset $R\subseteq \{1,\dots ,n\}$ there exists an (arithmetic) $k$-progression $A$ in $\{1,\dots ,\text{ac}(n,k)\}$ with $\{f(a):a\in A\}=R$. Determining the behaviour of the function $\text{ac}(n,k)$ is a previously unstudied problem. We use the first moment method to give an asymptotic upper bound for $\text{ac}(n,k)$ for the case $k=o(n^{1/5})$.