# rainbow

## Rainbow AP(4) in an almost equinumerous coloring ★★

Author(s): Conlon

\begin{problem} Do 4-colorings of $\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either $\lfloor p/4\rfloor$ or $\lceil p/4\rceil$? \end{problem}

Keywords: arithmetic progression; rainbow

## Long rainbow arithmetic progressions ★★

Author(s): Fox; Jungic; Mahdian; Nesetril; Radoicic

For $k\in \mathbb{N}$ let $T_k$ denote the minimal number $t\in \mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of $\{ 1,2,\ldots ,tn\}$ for every $n\in \mathbb{N}$

\begin{conjecture} For all $k\geq 3$, $T_k=\Theta (k^2)$. \end{conjecture}

Keywords: arithmetic progression; rainbow