A note on long rainbow arithmetic progressions

Jungi'{c} et al (2003) defined $T_k$ as the minimal number $t \in $\mathbb{N}$$suchthatthereisarainbowarithmeticprogressionoflength$k$ineveryequinumerous$t$-coloringof$[t n]$forevery$n \in $\mathbb{N}$$. They proved that for every $k\ge 3$, $\lfloor \frac{k^2}{4} \rfloor < T_{k} \leq \frac{k(k-1)^2}{2}$andconjecturedthat$T_{k} = \Theta(k^2)$. We prove for all $ϵ>0$ that $T_k=O(k^{5\mathrm{/}2+ϵ})$ using the K\H{o}v'{a}ri-S'{o}s-Tur'{a}n theorem and Wigert’s bound on the divisor function.