arithmetic progression

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

Concavity of van der Waerden numbers ★★

Author(s): Landman

For $k$ and $\ell$ positive integers, the (mixed) van der Waerden number $w(k,\ell)$ is the least positive integer $n$ such that every (red-blue)-coloring of $[1,n]$ admits either a $k$-term red arithmetic progression or an $\ell$-term blue arithmetic progression.

\begin{conjecture} For all $k$ and $\ell$ with $k \geq \ell$, $w(k,\ell) \geq w(k+1,\ell-1)$. \end{conjecture}

Keywords: arithmetic progression; van der Waerden

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