  Let the integers $1,\ldots,n$ be assigned colors. Szemer\'edi's theorem
implies that if there is a dense color class then there is an arithmetic
progression of length three in that color. We study the conditions on the color
classes forcing totally multicolored arithmetic progressions of length 3.
  Let $f(n)$ be the smallest integer $k$ such that there is a coloring of $\{1,
\ldots, n\}$ without totally multicolored arithmetic progressions of length
three and such that each color appears on at most $k$ integers. We provide an
exact value for $f(n)$ when $n$ is sufficiently large, and all extremal
colorings. In particular, we show that $f(n)= 8n/17 + O(1)$. This completely
answers a question of Alon, Caro and Tuza.
