If $W(k)$ is the least $N$ such that every two-colouring of $\{1, \dots, N\}$ contains a monochromatic $k$-term arithmetic progression, must $W(k+1) - W(k) \to \infty$?

Asserts that for any number of colors r and any progression length k, there always exists some number N large enough to guarantee a monochromatic arithmetic progression. In other words, the set monoAP_guarantee_set is non-empty. This is the fundamental existence result that allows for the definition of the van der Waerden numbers.