I am trying to develop bounds for the function B(k) where B(k) is defined as the least such positive integer so that whenever the set $\{1,2,\cdots B(k)\}$ is partitioned into two parts at least one part contains a set of order k, $\{x_1 < x_2 < \cdots x_k\}$, where the differences $x_i-x_{i-1}\in \{ a,b \}$ for some $a,b\in \mathbb{Z}^+$. Clearly $B_k$ exists since the van der Waerden theorem guarantees its existence.

What I am specifically interested to know is to whether someone has tackled this problem before and what results have been obtained in that connection.

Thanks