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 $k$?



Density in van der Waerden's theorem
They grow very fast. Denote by N(l) the largest integer such that we can color the numbers with two colors from 1 to N(l) without an l-long arithmetic progression. Now if you use this coloring for (k-1)/2-long sequences and put such colorings after each other, then any k-long arithmetic progression will be longer than N((k-1)/2), which means that even its difference will be at least N((k-1)/2)/k, which is HUGE.


