![](/files/happy5.png)
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
,
.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ k \geq \ell $](/files/tex/739b55f8a55e492f6a853fe336e2a04b05bd1182.png)
![$ w(k,\ell) \geq w(k+1,\ell-1) $](/files/tex/783e69162d75cf0eb2d5bd5a8d8b3410b04e5107.png)
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.