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