A note on the mixed van der Waerden number
Bull. Korean Math. Soc. 2021 Vol. 58, No. 6, 1341-1354
https://doi.org/10.4134/BKMS.b200718
Published online November 1, 2021
Printed November 30, 2021
Kai An Sim, Ta Sheng Tan, Kok Bin Wong
Sunway University; Universiti Malaya; Universiti Malaya
Abstract : Let $r \geq 2$, and let $ k_i \geq 2$ for $1 \leq i \leq r$. Mixed van der Waerden's theorem states that there exists a least positive integer $w= w(k_1, k_2, k_3, \dots, k_r;r)$ such that for any $n \geq w$, every $r$-colouring of $[1,n]$ admits a $k_i$-term arithmetic progression with colour $i$ for some $i \in [1,r]$. For $k \geq 3$ and $r \geq 2$, the mixed van der Waerden number $w(k,2,2, \dots, 2;r)$ is denoted by $w_2(k;r)$. B. Landman and A. Robertson \cite{vdw5} showed that for $k < r < \frac{3}{2} (k-1)$ and $r \geq 2k+2$, the inequality $w_2(k;r) \leq r(k-1)$ holds. In this note, we establish some results on $w_2(k;r)$ for $2 \leq r \leq k$.
Keywords : Mixed van der Waerden number, Ramsey theory on the integers
MSC numbers : Primary 05C57, 05C80
Supported by : http://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.b200718
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