Given positive integers k_1, k_2, there is an integer m such that given any partition {1,2, ..., m}=P_1 U P_2, there is an arithmetic progression of length k_1 in P_1 or k_2 consecutive integers in P_2. Let us denote the least m with this property by w_1(2; k_1, k_2). Some previously unknown van der Waerden type numbers (bold) are computed (Aug. 2008), by Tanbir Ahmed (ta_ahmed at cs dot concordia dot ca) using the wonderful machines at ConCoCO Research Laboratory van der Waerden type numbers known so far:
w_1(2; k_1, k_2)	=	Reference
w_1(2; 3, 2)	= 9	Brown, Landman, Robertson [1]
w_1(2; 3, 3)	= 23	Brown, Landman, Robertson [1]
w_1(2; 3, 4)	= 34	Brown, Landman, Robertson [1]
w_1(2; 3, 5)	= 73	Brown, Landman, Robertson [1]
w_1(2; 3, 6)	= 113	Brown, Landman, Robertson [1]
w_1(2; 3, 7)	= 193	Brown, Landman, Robertson [1]
w_1(2; 3, 8)	= 238	*
w_1(2; 4, 2)	= 18	*
w_1(2; 4, 3)	= 62	*
w_1(2; 4, 4)	= 229	*
w_1(2; 5, 2)	= 32	*
w_1(2; 6, 2)	= 60	*
w_1(2; 7, 2)	= 301	*
[1] Brown T., Landman B., Robertson A., Bounds on some van der Waerden numbers, Journal of Combinatorial Theory, Series A Volume 115, Issue 7 (2008) 1304-1309 Known van der Waerden numbers.