Consider a coloring of $\mathbb{N}$ with two colors. How many monochromatic arithmetic progressions of a fixed length $m$ (i.e. numbers of the form $a+nd$ are colored the same) are there in the subset $\{1,2, \cdots, k\}$?
I have counted this for a few small values of $k$ by hand and it seems very complicated to count this in general due to the large number of overlaps. Is it possible to generalize this to any number of colorings? I would also be interested in an asymptotic answer. Note: This relates to Van der Waarden's Theorem.