Claims

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1973_01_01_erdos: Erdős's 1973 observation that coloring n by the fractional part of n times a quadratic irrational bounds every monochromatic progression of difference d by a constant times d, so the best f is at most linear in d.

1980_11_01_beck: Beck's 1980 theorem gives, for every epsilon > 0, a two-coloring of the integers in which no monochromatic progression of difference d is longer than (1 + epsilon) log_2 d for large d; the best f is at most (1 + o(1)) log_2 d.

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