TY - JOUR
T1 - Monochromatic arithmetic progressions with large differences
AU - Brown, Tom C.
AU - Landman, Bruce M.
PY - 1999/8
Y1 - 1999/8
N2 - A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f : Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id : 0 ≤ i ≤ k - 1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.
AB - A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f : Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id : 0 ≤ i ≤ k - 1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.
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U2 - 10.1017/s0004972700033293
DO - 10.1017/s0004972700033293
M3 - Article
AN - SCOPUS:0033177589
SN - 0004-9727
VL - 60
SP - 21
EP - 35
JO - Bulletin of the Australian Mathematical Society
JF - Bulletin of the Australian Mathematical Society
IS - 1
ER -