= Alternating dyadic colouring excludes symmetric weighted pair sums
{title2=$c(n)=\lfloor\log_2 n\rfloor\pmod 2$}
Define a <finite colouring> of the positive <integers> by $c(n)=\lfloor\log_2 n\rfloor\pmod 2$. No increasing infinite <sequence> has all the numbers $x_i+2x_j$, $i\ne j$, in one <colour class>.
Write $\delta(y)=2^{\lfloor\log_2 y\rfloor+1}-y$. If $\delta$ is unbounded on the <sequence>, fix a term $x$ and choose a later term $y>2x$ with $\delta(y)>2x$. With $L=2^{\lfloor\log_2 y\rfloor}$, the numbers $y+2x$ and $2y+x$ lie respectively in the adjacent <dyadic intervals> $[L,2L)$ and $[2L,4L)$, so their colours differ. If $\delta$ is bounded by $D$ on the <sequence>, choose $x>2D$ and then $y>2x$. Now $y+2x=2L-\delta(y)+2x$ lies in $[2L,4L)$, whereas $2y+x=4L-2\delta(y)+x$ lies in $[4L,8L)$. Their colours again differ. The <finite colouring> therefore excludes the symmetric weighted-pair configuration.
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