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On a Variant of Van Der Waerden's Theorem
Published/Copyright:
May 31, 2010
Abstract
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 obtain an exponential lower bound on Q1(k) using probabilistic techniques and linear algebra.
Received: 2009-11-06
Published Online: 2010-05-31
Published in Print: 2010-May
© de Gruyter 2010
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Articles in the same Issue
- Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions
- Multiplicities of Integer Arrays
- Adjugates of Diophantine Quadruples
- A Binary Tree Representation for the 2-Adic Valuation of a Sequence Arising From a Rational Integral
- On a Variant of Van Der Waerden's Theorem
- Note on a Result of Haddad and Helou
- Sequences of Density ζ(k) – 1
- Square-Full Divisors of Square-Full Integers
- Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms
