Monochromatic Sequences Free of a Set of Specified Difference(s)

There is a minimum positive integer n such that every coloring of {1, 2, ..n} with r colors produces a monochromatic sequence of length k. Numbers in this monochromatic sequence should avoid a specified set of difference(s), {d}, with themselves. This showcase aims to explain how formulas were developed to find this minimum number for sets of difference {2} and differences {2, 2q} for all values of r, k.