On generalized van der Waerden triples

Abstract

Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k, r), such that any r-coloring of {1, 2, . . . , w(k, r)} must contain a monochromatic kterm arithmetic progression {x, x + d, x + 2d, . . . , x + (k − 1)d}. We investigate the following generalization of w(3, r). For fixed positive integers a and b with a ≤ b, define N (a, b; r) to be the least positive integer, if it exists, such that any r-coloring of {1, 2, . . . , N (a, b; r)} must contain a monochromatic set of the form {x, ax + d, bx + 2d}. We show that N (a, b; 2) exists if and only if b = 2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a, b), N (a, b; r) does not exist for r sufficiently large. We also give a result on sets of the form {x, ax + d, ax + 2d, . . . , ax + (k − 1)d}. N → N, the Ramsey function corresponding to the collection of arithmetic progressions {a, a + d, a + 2d, ..., a + (k − 1)d} with the property that d ≥ f (a). The Ramsey functions for other "substitutes" for the set of arithmetic progressions were studied in [2], [4], and [5]. In this paper we consider a new generalization of w(k, r). To help describe this generalization, we begin with three definitions.of three natural numbers is called an (a, b)triple if there exist natural numbers x and d such that S = {x, ax + d, bx + 2d}. Definition 1.2: Fix 1 ≤ a ≤ b. Define N(a, b; r) to be the least positive integer, if it exists, such that any r-coloring of [1, N(a, b; r)] must contain a monochromatic (a, b)triple. Definition 1.3: Fix 1 ≤ a ≤ b. Define (a, b) to be regular if N(a, b; r) exists for all positive integers r. If (a, b) is not regular, the degree of regularity of (a, b) is the largest r such that N(a, b; r) exists. Denote this by dor(a, b).We note here that N(1, 1; r) is the van der Waerden number w(3, r) so that N(a, b; r) is a generalization of w(3, r), and obviously (1, 1) is regular. We now discuss the sections which follow.In Section 2 we consider r = 2. We show that, except for the case in which b = 2a, N(a, b; 2) does exist; we also find upper and lower bounds on N(a, b; 2) (for b = 2a). For certain pairs (a, b), we obtain stronger bounds; in particular, we use a result of [1] to deal with N(a, 2a − 1; 2) (when a = 1 this is just w(3, 2)). In Section 3 we establish that (a, b) is not regular for a rather large class of pairs (a, b), and give an upper bound (for these pairs) on the degree of regularity. We then give lower bounds on N(a, b; r) for all 1 ≤ a ≤ b and r > 2. In Section 4 we make some observations about monochromatic sets of the form {x, ax + d, ax + 2d, . . . , ax + (k − 1)d} for a ≥ 1. We establish that for a > 1 and k sufficiently large (dependent upon a), we can 4-color the natural numbers so that no monochromatic such k-set exists (this is in contrast to van der Waerden's theorem which says that there are arbitrarily long monochromatic arithmetic progressions in any r-coloring of the natural numbers). Using Tw...

On the degree of regularity of generalized van der Waerden triples

“…In Section 3 we show this conjecture to be false. We also obtain upper bounds on dor(a, b) for all (a, b) = (1, 1), which improve upon the results of [3], and provide an alternate proof that (1,1) is the only regular triple.…”

“…It also uses the following lemma. In [3] it was shown that dor(1, 3) ≤ 3, dor(2, 3) = 2, and dor(2, 2) ≤ 5. By Lemma 1, these three facts cover Cases (i), (ii), and (iii), respectively.…”

“…van der Waerden [5] proved that for any positive integers k and r, there is a positive integer w(k, r) such that any r-coloring of {1, 2, ..., w(k, r)} must admit a monochromatic k-term arithmetic progression. In [3], a generalization of van der Waerden's theorem for 3-term arithmetic progressions was investigated. Namely, for integers 1 ≤ a ≤ b, define an (a, b)-triple to be any 3-term sequence of the form (x, ax + d, bx + 2d), where x, d are positive integers.…”

“…In [3], it is shown that for a wide class of pairs (a, b) = (1, 1), (a, b) is not regular, i.e., dor(a, b) < ∞, and its authors conjectured that, in fact, (1, 1) is the only regular pair. In Section 2 we confirm this conjecture.…”

Disjunctive Rado numbers

“…Recently, several other problems related to Schur numbers and Rado numbers have been considered [1][2][3]6,7,13,17,19]. In 2001, Robertson and Schaal introduced the concept of off-diagonal Rado numbers (or off-diagonal generalized Schur numbers) [18].…”