The mixed van der Waerden number w(k1,. .. , kr; r) is the least positive integer n such that every r-colouring of [1, n] admits a monochromatic arithmetic progression of length ki, for at least one i. We denote by w2(k; r) the case in which k1 = • • • = kr−1 = 2 and kr = k. For k ≤ r, we give sharp upper and lower bounds for w2(k; r), indicating also cases when these bounds are achieved. We determine exact values in the cases where (k, r) ∈ {(p, p), (p, p + 1), (p + 1, p + 1)} and give bounds in the cases where (k, r) ∈ {(p, p + 2), (p + 2, p + 2)}, for primes p. We provide a table of values for the cases k ≤ r with 3 ≤ k ≤ 10 and for several values of r, correcting some known values.
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