Color the positive integers using just two colors. By van der Waerden's theorem, we can find a $k$-term arithmetic progression as long as we consider a long interval.

I imagine it is possible to find a $k$-term arithmetic progression so that the terms in the progression have minimal gaps by possibly taking an even longer interval (of some fixed size depending only on $k$). If so, how do these minimal gaps behave as a function of $k$?