Arithmetic Progressions in Sequences with Bounded Gaps

Let G(k;r) denote the smallest positive integer g such that if 1 = a1;a2; : : : ;ag is a strictly increasing sequence of integers with bounded gaps a j+1 a j r, 1 j g 1, then fa1;a2; : : : ;agg contains a k-term arithmetic progression. It is shown that G(k;2) > q k 1 2 4 3 k 1 2 , G(k;3) > 2 k 2 ek (1+ o(1)), G(k;2r 1)> r k 2 ek (1+o(1)), r 2. For positive integers k, r, the van der Waerden number W (k;r) is the least integer such that if w W (k;r), then any partition of [1;w] into r parts has a part that contains a k-term arithmetic progression. The celebrated theorem of van der Waerden [4] proves the existence of W (k;r). The best known upper bound for W (k;2) is enormous whereas the best known lower bound for W (k;2) (see [1]) is W (k;2)> 2k 2ek (1+o(1)) (1) where e is the base of the natural logarithm. Let G(k;r) denote the smallest positive integer g such that if 1 = a1;a2; : : : ;ag is a strictly increasing sequence of integers with bounded gaps a j+1 a j r, 1 j g 1, then fa1;a2; : : : ;agg contains a kterm arithmetic progression. In [3], Rabung notes that van der Waerden’s theorem implies the existence of G(k;r) for all k, r and conversely. Nathanson makes the following quantitative connection between W (k;r) and G(k;r) [2, Theorem 4]: G(k;r) W (k;r) G((k 1)r+1;2r 1): (2) In particular, W (k;2) G(2k 1;3), which suggests that it is no easier to find a reasonable upper bound for G(k;3) than it is for W (k;2). However, G(k;2) “escapes” Nathanson’s inequalities in the sense that an upper bound for G(k;2) does not immediately give an upper bound for W (k;2). Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6. [email protected]. †Department of Mathematics and Statistics, Okanagan University College, 3333 College Way, Kelowna, B.C. Canada V1V 1V7 [email protected]. ‡Both authors gratefully acknowledge the support of the National Science and Engineering Research Council of Canada.