For $k\in \mathbb{N}$ let $T_k$ denote the minimal number $t\in \mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of $\{ 1,2,\ldots ,tn\}$ for every $n\in \mathbb{N}$
\begin{conjecture} For all $k\geq 3$, $T_k=\Theta (k^2)$. \end{conjecture}