A NOTE ON THE MIXED VAN DER WAERDEN NUMBER
A NOTE ON THE MIXED VAN DER WAERDEN NUMBER
- 대한수학회
- Bulletin of the Korean Mathematical Society
- Vol.58 No.6
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2021.011341 - 1354 (14 pages)
- 0
Let r ≥ 2, and let k<sub>i</sub> ≥ 2 for 1 ≤ i ≤ r. Mixed van der Waerden's theorem states that there exists a least positive integer w = w(k<sub>1</sub>, k<sub>2</sub>, k<sub>3</sub>, …, k<sub>r</sub>; r) such that for any n ≥ w, every r-colouring of [1, n] admits a k<sub>i</sub>-term arithmetic progression with colour i for some i ∈ [1, r]. For k ≥ 3 and r ≥ 2, the mixed van der Waerden number w(k, 2, 2, …, 2; r) is denoted by w<sub>2</sub>(k; r). B. Landman and A. Robertson [9] showed that for k < r < $\frac{3}{2}$(k - 1) and r ≥ 2k + 2, the inequality w<sub>2</sub>(k; r) ≤ r(k - 1) holds. In this note, we establish some results on w<sub>2</sub>(k; r) for 2 ≤ r ≤ k.
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