%0 Journal Article %T On rainbow 4-term arithmetic progressions %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Shirdareh Haghighi, M. H. %A Salehi Nowbandegani, P. %D 2012 %\ 09/15/2012 %V 37 %N No. 3 %P 33-37 %! On rainbow 4-term arithmetic progressions %K Rainbow arithmetic progression %K 4-term arithmetic progression %K AP(4) %K AP($k$) %R %X {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 $k=3$.} %U http://bims.iranjournals.ir/article_350_89e63dba5b5df3fefe16dd3a01425b8e.pdf