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 $k = o(n^{ 1/{ 5}})$.
Categories: math.CO