Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions
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 ark (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 …
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 ark (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 …
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.
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