# Long rainbow arithmetic progressions

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

**Conjecture**For all , .

A -coloring of is equinumerous if each color is used times. An arithmetic progression is rainbow if it does not containt two terms of the same color.

In [JLMNR] it was proved that .

It is known that ([AF], [JR]) and ([CJR]). It is not hard to show that for all ([AF]).

## Bibliography

[AF] Maria Axenovich, Dmitri Fon-Der-Flaass: On rainbow arithmetic progressions, Electronic Journal of Combinatorics, 11, (2004), R1.

[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254

*[JLMNR] Veselin Jungic, Jacob Licht (Fox), Mohammad Mahdian, Jaroslav Nesetril, Rados Radoicic : Rainbow arithmetic progressions and anti-Ramsey results, Combinatorics, Probability, and Computing - Special Issue on Ramsey Theory, 12, (2003), 599--620.

[JNR] Veselin Jungic, Jaroslav Nesetril, Rados Radoicic: Rainbow Ramsey theory, Integers, The Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference 2003 in Honor of Tom Brown, 5(2), (2005), A9.

[JR] Veselin Jungic, Rados Radoicic : Rainbow 3-term arithmetic progressions, Integers, The Electronic Journal of Combinatorial Number Theory, 3, (2003), A18.

* indicates original appearance(s) of problem.

## Is this unsolved?

Looks like a nice problem, but is it still unsolved?