Concavity of van der Waerden numbers

Importance: Medium ✭✭
Author(s): Landman, Bruce M.
Recomm. for undergrads: no
Posted by: Bruce Landman
on: June 21st, 2007

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.

Conjecture   For all $ k $ and $ \ell $ with $ k \geq \ell $, $ w(k,\ell) \geq w(k+1,\ell-1) $.

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


*[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.