Let $\ac(n,k)$ denote the smallest positive integer

with the property that there exists an $n$-colouring $f$ of

$\{1,\dots,\ac(n,k)\}$ such that for every $k$-subset

$R \subseteq \{1, \dots, n\}$ there exists an (arithmetic)

$k$\nobreakdash-progression $A$ in $\{1,\dots,\ac(n,k)\}$

with $\{f(a) : a \in A\} = R$.

Determining the behaviour of the function $\ac(n,k)$

is a previously unstudied problem.

We use the first moment method to give

an asymptotic upper bound for $\ac(n,k)$ for the case $k = o(n^{1/{5}})$.