On rainbow 4-term arithmetic progressions

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$ also exists for odd $n>1$. We conclude that for nonnegative integers $kgeq 3$ and $n > 1$, every equinumerous $k$-coloring of $[kn]$ contains a rainbow AP$(k)$ if and only if $k=3$.}