A note on long rainbow arithmetic progressions

Jungi\'{c} et al (2003) defined $T_{k}$ as the minimal number $t \in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[t n]$ for every $n \in \mathbb{N}$. They proved that for every $k \geq 3$, $\lfloor \frac{k^2}{4} \rfloor < T_{k} \leq \frac{k(k-1)^2}{2}$ and conjectured that $T_{k} = \Theta(k^2)$. We prove for all $\epsilon > 0$ that $T_{k} = O(k^{5/2+\epsilon})$ using the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem and Wigert's bound on the divisor function.