arithmetic progression

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

Author(s): Conlon

Problem   Do 4-colorings of , for a large prime, always contain a rainbow if each of the color classes is of size of either or ?

Keywords: arithmetic progression; rainbow

Long rainbow arithmetic progressions ★★

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

For let denote the minimal number such that there is a rainbow in every equinumerous -coloring of for every

Conjecture   For all , .

Keywords: arithmetic progression; rainbow

Concavity of van der Waerden numbers ★★

Author(s): Landman

For and positive integers, the (mixed) van der Waerden number is the least positive integer such that every (red-blue)-coloring of admits either a -term red arithmetic progression or an -term blue arithmetic progression.

Conjecture   For all and with , .

Keywords: arithmetic progression; van der Waerden