Weak Arithmetic Progressions

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?

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Why can't I see the whole question? :S –  Shahab Jun 5 '12 at 11:15
When I click edit I can see the whole question but otherwise I am seeing only one & a half sentences. Does anyone know why is this happening? –  Shahab Jun 5 '12 at 11:20
You have to leave a space after any < character, otherwise markdown will interpret it as a start of an HTML tag. Also, you need to double backslashes in commands like \{, \}, \\. –  Emil Jeřábek Jun 5 '12 at 11:24
Ok, thanks for the information. –  Shahab Jun 5 '12 at 11:38
Is WAP also known as "bounded gaps"? If so, search that term for lots of info. –  Gerald Edgar Jun 5 '12 at 12:01