TY - JOUR ID - 350 TI - On rainbow 4-term arithmetic progressions JO - Bulletin of the Iranian Mathematical Society JA - BIMS LA - en SN - 1017-060X AU - Shirdareh Haghighi, M. H. AU - Salehi Nowbandegani, P. AD - Y1 - 2012 PY - 2012 VL - 37 IS - No. 3 SP - 33 EP - 37 KW - Rainbow arithmetic progression KW - 4-term arithmetic progression KW - AP(4) KW - AP($k$) DO - N2 - {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 $k=3$.} UR - http://bims.iranjournals.ir/article_350.html L1 - http://bims.iranjournals.ir/article_350_89e63dba5b5df3fefe16dd3a01425b8e.pdf ER -