Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON THE MIXED VAN DER WAERDEN NUMBER

  • Sim, Kai An (School of Mathematical Sciences Sunway University) ;
  • Tan, Ta Sheng (Institute of Mathematical Sciences Universiti Malaya) ;
  • Wong, Kok Bin (Institute of Mathematical Sciences Universiti Malaya)
  • Received : 2020.08.24
  • Accepted : 2021.08.19
  • Published : 2021.11.30
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Abstract

Let r ≥ 2, and let ki ≥ 2 for 1 ≤ i ≤ r. Mixed van der Waerden's theorem states that there exists a least positive integer w = w(k1, k2, k3, …, kr; r) such that for any n ≥ w, every r-colouring of [1, n] admits a ki-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 w2(k; r). B. Landman and A. Robertson [9] showed that for k < r < $\frac{3}{2}$(k - 1) and r ≥ 2k + 2, the inequality w2(k; r) ≤ r(k - 1) holds. In this note, we establish some results on w2(k; r) for 2 ≤ r ≤ k.

Keywords

Acknowledgement

This research was funded by the Fundamental Research Grant Scheme (FRGS) grant number FRGS/1/2020/STG06/SYUC/03/1 by Malaysia Ministry of Higher Education.

References

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