The 2-color relative linear Van der Waerden numbers

Byeong Moon Kim; Yoomi Rho

doi:10.1016/j.crma.2007.06.025

Combinatorics

The 2-color relative linear Van der Waerden numbers

Presented by: Christophe Soulé

Byeong Moon Kim ¹; Yoomi Rho ²

¹ Department of Mathematics, Kangnung National University, Kangnung 210-702, Republic of Korea
² Department of Mathematics, University of Incheon, Incheon 402-749, Republic of Korea

Comptes Rendus. Mathématique, Volume 345 (2007) no. 4, pp. 183-186.

Abstract
Résumé

We define the r-color relative linear van der Waerden numbers for a positive integer r as generalizations of the polynomial van der Waerden numbers of linear polynomials. Especially we express a sharp upper bound of the 2-color relative linear van der Waerden number $R f_{2} (u_{1}, u_{2}, \dots, u_{m} : s_{1}, s_{2}, \dots, s_{k})$ in terms of a $(k + 1)$ -color polynomial van der Waerden number for positive integers m, k, $u_{1}, u_{2}, \dots, u_{m}$ , $s_{1}, s_{2}, \dots, s_{k}$ . As a result, we find this upper bound for some instances of m, k, $u_{1}, u_{2}, \dots, u_{m}$ , $s_{1}, s_{2}, \dots, s_{k}$ for which the $(k + 1)$ -color polynomial van der Waerden numbers are obtained.

Nous définissons les nombres de van der Waerden linéaires relatifs r-colorés pour un entier strictement positif r qui sont des généralisations des nombres polynomiaux de van der Waerden de polynôme linéaires. En particulier nous donnons, pour $r = 2$ , la borne supérieure de ces nombres $R f_{2} (u_{1}, u_{2}, \dots, u_{m} : s_{1}, s_{2}, \dots, s_{k})$ en termes d'un nombre de van der Waerden polynomial $(k + 1)$ -coloré pour les entiers strictement positifs, m, k, $u_{1}, u_{2}, \dots, u_{m}$ , $s_{1}, s_{2}, \dots, s_{k}$ . Comme conséquence, nous obtenons explicitement cette borne supérieure pour certaines valeurs de ces entiers pour lesquels les nombres polynomiaux de van der Waerden $(k + 1)$ -colorés peuvent être calculés.

Received: 2007-01-16
Accepted: 2007-06-26
Published online: 2007-08-09

DOI: 10.1016/j.crma.2007.06.025

Author's affiliations:

Byeong Moon Kim ¹; Yoomi Rho ²

¹ Department of Mathematics, Kangnung National University, Kangnung 210-702, Republic of Korea
² Department of Mathematics, University of Incheon, Incheon 402-749, Republic of Korea

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     author = {Byeong Moon Kim and Yoomi Rho},
     title = {The 2-color relative linear {Van} der {Waerden} numbers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {183--186},
     publisher = {Elsevier},
     volume = {345},
     number = {4},
     year = {2007},
     doi = {10.1016/j.crma.2007.06.025},
     language = {en},
}

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Byeong Moon Kim; Yoomi Rho. The 2-color relative linear Van der Waerden numbers. Comptes Rendus. Mathématique, Volume 345 (2007) no. 4, pp. 183-186. doi : 10.1016/j.crma.2007.06.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.025/

References
Cited by

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[5] B.M. Kim; Y. Rho Van der Waerden's theorem on homothetic copies of ${1, 1 + s, 1 + s + t}$ (preprint, arxiv:) | arXiv

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Cited by Sources:

^⁎ This work is supported by the University of Incheon research grant in 2005.

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