Long rainbow arithmetic progressions

Define T_k as the minimal t ∈ N for which there is a rainbow arithmetic progression of length k in every equinumerous t-coloring of [tn] for all n ∈ N. Jungić, Licht (Fox), Mahdian, Ne˘set˘ril and Radoi˘cić provedthat⌊^k2 4^⌋≤Tk ≤ k(k − 1)²/2. We almost close the gap between the upper and lower bounds by proving that T_k ≤ k²e^{(ln lnk)2(1+o(1))}. Conlon, Fox and Sudakov have independently shown a stronger statement that T_k = O(k² log k).