Open Access
2018 Some results on a class of mixed van der Waerden numbers
Kaushik Maran, Sai Praneeth Reddy, Dravyansh Sharma, Amitabha Tripathi
Rocky Mountain J. Math. 48(3): 885-904 (2018). DOI: 10.1216/RMJ-2018-48-3-885


The mixed van der Waerden number ${w}(k_1, \ldots ,k_r;r)$ is the least positive integer $n$ such that every $r$-coloring of $[1,n]$ admits a monochromatic arithmetic progression of length $k_i$, for at least one $i$. We denote by ${w}_2(k;r)$ the case in which $k_1=\cdots =k_{r-1}=2$ and $k_r=k$. For $k \le r$, we give upper and lower bounds for ${w}_2(k;r)$, also indicating cases when these bounds are achieved. We determine exact values in the cases where $(k,r) \in \{(p,p),(p,p+1),(p+1,p+1)\}$ and give bounds in the cases where $(k,r) \in \{(p,p+2),(p+2,p+2)\}$, for primes $p$. We provide a table of values for the cases $k \le r$ with $3 \le k \le 10$ and for several values of $r$, correcting some known values.


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Kaushik Maran. Sai Praneeth Reddy. Dravyansh Sharma. Amitabha Tripathi. "Some results on a class of mixed van der Waerden numbers." Rocky Mountain J. Math. 48 (3) 885 - 904, 2018.


Published: 2018
First available in Project Euclid: 2 August 2018

zbMATH: 06917352
MathSciNet: MR3835577
Digital Object Identifier: 10.1216/RMJ-2018-48-3-885

Primary: 05D10
Secondary: 05C55

Keywords: $r$-coloring , arithmetic progression , complete residue system , valid coloring

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

Vol.48 • No. 3 • 2018
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