= Dyadic block obstruction to infinite arithmetic progressions
{title2=$\chi(n)=\lfloor\log_2 n\rfloor\bmod2$}
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>.
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