I am studying a special type of a sequence on the naturals which I am calling a weak arithmetic progression.

Formally I call a k-sequence $x_1< x_2 \cdots< x_k$ a weak arithmetic progression (WAP) if $\exists d\in \mathbb{N}$ such that $x_{i+1}-x_i\in\{1,d\}$. Now supposing all positive integers have been partitioned into two parts is there an upper bound on how many times a segment of $t$ consecutive numbers ($0\leq t\leq \frac{k-1}{2}$) in one part can occur without a WAP occurring in any one part?