# Concavity of van der Waerden numbers

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 , .

The conjecture was stated in 2000 and published 2003 [LR] and 2007 [KL].

## Bibliography

*[BL] Bruce Landman and Aaron Robertson, *Ramsey Theory on the Integers*, American Mathematical Society, Providence, Rhode Island, 2003.

[KL] Abdollah Khodkar and Bruce Landman, Recent progress in Ramsey theory on the integers, in *Combinatorial Number Theory*, 305-313, de Gruyter, Berlin, 2007.

* indicates original appearance(s) of problem.