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van der Waerden
Concavity of van der Waerden numbers ★★
Author(s): Landman
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)
Keywords: arithmetic progression; van der Waerden
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