Van der Waerden numbers
NEW: Visualize your own certificate.
For the latest description of our method see Improving the
odds.
r \ k 
3 
4 
5 
6 
7 
8 
9 
2 
9 
35 
178 
1132[3] 
> 3703[4] 
> 7484[4] 
>27113[4] 
3 
27 
> 292[4] 
> 1209[?] 
> 8886[4] 
> 43855[4] 
>238400[4*] 

4 
76 
> 1048[4] 
> 10437[4] 
> 90306[4] 
>387967[4*] 


5 
>125[1] 
>2254[4] 
>24045[4] 
>246956[4*] 



6 
>207[4] 
>9778[4] 
>56693[4*] 
>600486[4*] 



Table 1: Current best known lower bounds to Van der Waerden numbers
* This lower bound can be constructed using the method described in the paper, but due to lack of computational power these results are
not mentioned.
Table 2: Improved best known lower bounds to Van der Waerden numbers
Of all best known lower bounds there exists a certificate that has some symmetric properties.
To show these symmetries one can use the visualization technique described in [2]:
When rpartitioning is involved we use r directions in the plane, where the angle
between two consecutive directions is 360 / r . Starting from the beginning of the certicate
a line segment is drawn in the direction associated with the subset containing number 1.
From the endpoint of that line segment a line segment with equal length is drawn in the direction
associated with the subset containing number 2. This process is repeated up to number n of the
certicate. The line segments are gradually colored from red to blue to green and back to red.
This visualization is only applicable for r > 2.
r \ k 
3 
4 
5 
6 
3 
W(3,3) > 26 
W(3,4) > 292 
W(3,5) > 2173 
W(3,6) > 11191 
4 
W(4,3) > 75 
W(4,4) > 1048 
W(4,5) > 17705 
W(4,6) > 91331 
5 
W(5,3) > 170 
W(5,4) > 2254 
W(5,5) > 98740 
W(5,6) > 540025 
Table 3: Visualisations of a certificate of the best known lower bounds to Van der Waerden numbers
[1] M.R. Dransfield, L. Liu, V. Marek, M. Truszczynski.
Satisfiability and Computing van der Waerden numbers.
The Electronic Journal of Combinatorics, vol. 11 (1) (2004).
[2] P. Herwig, M.J.H. Heule, M. van Lambalgen, and H. van Maaren.
A new method to construct lower bounds for Van der Waerden numbers.
The Electronic Journal of Combinatorics 14 (2007) #R6. [ .pdf ]
[3] M. Kouril and J.L. Paul.
The Van der Waerden Number W(2,6) is 1132.
Experimental Mathematics (to appear).
[4] J.R. Rabung.
Some Prgrogressionfree Partitions Constructed Using Folkman's Method.
Canadian Mathematical Bulletin, vol. 22 (1979) 8791.
