Geodesic
A new method to construct lower bounds for van der Waerden numbers
The electronic journal of combinatorics, Tome 14 (2007)
Cet article a éte moissonné depuis la source The Electronic Journal of Combinatorics website

Voir la notice de l'article

We present the Cyclic Zipper Method, a procedure to construct lower bounds for Van der Waerden numbers. Using this method we improved seven lower bounds. For natural numbers $r$, $k$ and $n$ a Van der Waerden certificate $W(r,k,n)$ is a partition of $\{1, \ldots, n\}$ into $r$ subsets, such that none of them contains an arithmetic progression of length $k$ (or larger). Van der Waerden showed that given $r$ and $k$, a smallest $n$ exists - the Van der Waerden number $W(r,k)$ - for which no certificate $W(r,k,n)$ exists. In this paper we investigate Van der Waerden certificates which have certain symmetrical and repetitive properties. Surprisingly, it shows that many Van der Waerden certificates, which must avoid repetitions in terms of arithmetic progressions, reveal strong regularities with respect to several other criteria. The Cyclic Zipper Method exploits these regularities. To illustrate these regularities, two techniques are introduced to visualize certificates.
DOI : 10.37236/925
Classification : 05D10
Mots-clés : van der Waerden numbers, arithmetic progressions 
@article{10_37236_925,
     author = {P. R. Herwig and M.J.H. Heule and P. M. van Lambalgen and H. van Maaren},
     title = {A new method to construct lower bounds for van der {Waerden} numbers},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/925},
     zbl = {1115.05092},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/925/}
}
TY  - JOUR
AU  - P. R. Herwig
AU  - M.J.H. Heule
AU  - P. M. van Lambalgen
AU  - H. van Maaren
TI  - A new method to construct lower bounds for van der Waerden numbers
JO  - The electronic journal of combinatorics
PY  - 2007
VL  - 14
UR  - http://geodesic.mathdoc.fr/articles/10.37236/925/
DO  - 10.37236/925
ID  - 10_37236_925
ER  - 
%0 Journal Article
%A P. R. Herwig
%A M.J.H. Heule
%A P. M. van Lambalgen
%A H. van Maaren
%T A new method to construct lower bounds for van der Waerden numbers
%J The electronic journal of combinatorics
%D 2007
%V 14
%U http://geodesic.mathdoc.fr/articles/10.37236/925/
%R 10.37236/925
%F 10_37236_925
P. R. Herwig; M.J.H. Heule; P. M. van Lambalgen; H. van Maaren. A new method to construct lower bounds for van der Waerden numbers. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/925

Cité par Sources :