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 r k ( n ) denote the minimum number of colour required so that there exists a r k ( n ) -colouring of [ 1 , n ] that avoids any monochromatic k -term arithmetic progression. In this paper, we give necessary and sufficient conditions for r k ( n + 1 ) = r k ( n ) . We also show that r k ( n ) = 2 for all k ≤ n ≤ 2 ( k − 1 ) 2 and give an upper bound for r p ( p m ) for any prime p ≥ 3 and integer m ≥ 2 .