  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 $k = o(n^{1/{5}})$.
