arithmetic progression

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

Author(s): Conlon

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 $\lceil p/4\rceil$ ?

Keywords: arithmetic progression; rainbow

Posted by vjungic
updated September 1st, 2007

1 comment

Combinatorics

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 $n\in \mathbb{N}$

Conjecture For all $k\geq 3$ , $T_k=\Theta (k^2)$ .

Keywords: arithmetic progression; rainbow

Posted by vjungic
updated May 12th, 2010

1 comment

Combinatorics » Ramsey Theory

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.