Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X37No. 320120915On rainbow 4-term arithmetic progressions3337350ENM. H.Shirdareh HaghighiP.Salehi NowbandeganiJournal Article20090826{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 $k=3$.}http://bims.iranjournals.ir/article_350_89e63dba5b5df3fefe16dd3a01425b8e.pdf