Dyadic block obstruction to infinite arithmetic progressions ($\chi (n)=\lfloor {\log }_{2}n\rfloor \mathrm{mod}2$)

Colour each interval $[2^j,2^{j+1})\cap \mathbb{N}$ by the parity of $j$. Every infinite arithmetic progression of positive common difference d meets every sufficiently large such interval, since the interval has length greater than d. Both colours occur infinitely often. Thus neither colour class contains an infinite arithmetic progression, despite the finite-length conclusion of the Van der Waerden theorem.