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