On Avoiding Arithmetic Progressions Whose Common Differences Belong to a Given Small Set

It is well-known that if \(D\) is any finite set of integers, then there is an \(n\) large enough so that there exists a 2-coloring of the positive integers that avoids any monochromatic \(n\)-term arithmetic progressions whose common differences belong to \(D\).
If \(\vec{d} = (d_1, \ldots, d_k)\) and \(\vec{n} = (n_1, \ldots, n_k)\) are \(k\)-tuples of positive integers, denote by \(f_{\vec{d}}(\vec{n})\) the least positive integer \(N\), if it exists, such that for every 2-coloring of \([1, N]\) there is, for some \(i\), a monochromatic \(n_i\)-term arithmetic progression with common difference \(d_i\).
This paper looks at the problem of determining when \(f_{\vec{d}}(\vec{n})\) exists, and its value when it does exist, for \(k \leq 3\).
A complete answer is given for \(k = 2\).
A partial answer is given for \(k = 3\), including the fact that for all ordered triples \(\vec{d}\), \(f_{\vec{d}}(4, 4, 4)\) does not exist.