Abstract
By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.
Highlights
Let N be the set of positive integer, i.e., N = {1, 2, 3, . . . }
In an effort to investigate the van der Waerden theorem, Erdős and Turán [7] approached the problem in the opposite direction: Given a positive integer n, what is the maximum size of a subset of [1, n] that does not contain an arithmetic progression of length k? They define the following
We give an upper bound for r p for any prime p ≥ 3 and integer m ≥ 2 (Theorem 5)
Summary
N} that avoids three-term monotone arithmetic progressions. Davis et al [1] and Sidorenko [3] showed that there is arrangement of N that avoids three-term monotone arithmetic progressions. They [1] showed that there exists an arrangement of N that avoid five-term monotone arithmetic progressions.
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