“…where the first equality holds by Fact 3, and the last equality holds because g |G| = id, for every group element g ∈ G. Since ̺ |G| has L columns equal to the identity substitution distributed in arithmetic progression of difference d, any fixed point of ̺ has a monochromatic arithmetic progression of difference d and length at least L. This completes the proof for k = 1. This proof extends to every positive integer k because, since ̺ has a column which is equal to the identity substitution (the leftmost column), the group generated by the columns of ̺ is equal to the group generated by the columns of ̺ k [9]. This means for a fixed k 1, one can take ̺ k|G| and construct d (k) , this time with 0 m L k − 1.…”