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Designate by
W(
k;
t
0,
t
1,…,
t
k−1
) the smallest number
m, such that if all the integers in the set {1,2,…,
m} are partitioned into classes
V
0∪
V
1∪⋯∪
V
k-1
in any way, there will always exist a class
V
1 containing an arithmetic progression of
t
1 + 1 terms. This paper presents some new values for such numbers, together with the structures of the partitions of {1,2,…,
m - 1} which avoid arithmetic progressions of
t
1 + 1 terms. These results were obtained by a reasonably sophisticated computer search algorithm. Results of particular interest are that
W(4;2,2,2,2) = 76 and
W(2;4,4) = 178. The fact that
W(2;4,4) = 178 independently duplicates the recently published finding of R.S. Stevens and R. Shantaram [13].