# arithmetic progression

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

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**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 ?

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## Long rainbow arithmetic progressions ★★

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

**Conjecture**For all , .

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## Concavity of van der Waerden numbers ★★

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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 , .

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