For positive integers s and k1, k2, . . . , ks, let w(k1, k2, . . . , ks) be the minimum integer n such that any s-coloring {1, 2, . . . , n} → {1, 2, . . . , s} admits a ki-term arithmetic progression of color i for some i, 1 ≤ i ≤ s. In the case when k1 = k2 = · · · = ks = k we simply write w(k; s). That such a minimum integer exists follows from van der Waerden’s theorem on arithmetic progre...

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three-term the sense that all sufficiently large subsets at least one such progression. We prove there exists a positive constant c and set does progression, property for every subset , contains nontrivial three term derive this from more general quantitative Roth-type the...

Given positive integers n and k, a k-term quasi-progression of diameter n is a sequence (x1, x2, ..., xk) such that d ≤ xj+1−xj ≤ d+n, 1 ≤ j ≤ k−1, for some positive integer d. Thus an arithmetic progression is a quasi-progression of diameter 0. Let Qn(k) denote the least integer for which every coloring of {1, 2, ..., Qn(k)} yields a monochromatic k-term quasi-progression of diameter n. We obt...

Let N denote the set of all nonnegative integers. Let k ≥ 3 be an integer and A0 = {a1, . . . , at} (a1 < . . . < at) be a nonnegative set which does not contain an arithmetic progression of length k. We denote A = {a1, a2, . . . } defined by the following greedy algorithm: if l ≥ t and a1, . . . , al have already been defined, then al+1 is the smallest integer a > al such that {a1, . . . , al}...