Lower Bounds on some certain van der Waerden Functions

For positive integers r and k(1), k(2), ..., k(r), the vander Waerden number W(k(1), k(2), ... , k(r); r) is the minimum integer N such that whenever set {1, 2, ..., N} is partitioned into r sets S-1, S-2, ..., S-r, there is a k(i)-term arithmetic progression contained in Si for some i. This paper establishes an asymptotic lower bound for W(k, m; 2) for fixed m >= 3 which improves the result of T.C. Brown et al's in [Bounds on some van der Waerden numbers. J. Combin. Theory, Ser.A 115 (2008), 1304-1309]. Some lower bounds on certain van der Waerden-like functions are also proposed.